diff --git a/src/MacroExpander.js b/src/MacroExpander.js
index 01d703be..acf2fbc1 100644
--- a/src/MacroExpander.js
+++ b/src/MacroExpander.js
@@ -4,6 +4,8 @@
  * until only non-macro tokens remain.
  */
 
+import functions from "./functions";
+import symbols from "./symbols";
 import Lexer from "./Lexer";
 import {Token} from "./Token";
 import type {Mode} from "./types";
@@ -15,6 +17,16 @@ import type {MacroContextInterface, MacroDefinition, MacroExpansion}
     from "./macros";
 import type Settings from "./Settings";
 
+// List of commands that act like macros but aren't defined as a macro,
+// function, or symbol.  Used in `isDefined`.
+export const implicitCommands = {
+    "\\relax": true,     // MacroExpander.js
+    "^": true,           // Parser.js
+    "_": true,           // Parser.js
+    "\\limits": true,    // Parser.js
+    "\\nolimits": true,  // Parser.js
+};
+
 export default class MacroExpander implements MacroContextInterface {
     maxExpand: number;
     lexer: Lexer;
@@ -316,5 +328,19 @@ export default class MacroExpander implements MacroContextInterface {
 
         return expansion;
     }
+
+    /**
+     * Determine whether a command is currently "defined" (has some
+     * functionality), meaning that it's a macro (in the current group),
+     * a function, a symbol, or one of the special commands listed in
+     * `implicitCommands`.
+     */
+    isDefined(name: string): boolean {
+        return this.macros.has(name) ||
+            functions.hasOwnProperty(name) ||
+            symbols.math.hasOwnProperty(name) ||
+            symbols.text.hasOwnProperty(name) ||
+            implicitCommands.hasOwnProperty(name);
+    }
 }
 
diff --git a/src/Namespace.js b/src/Namespace.js
index 74d59d56..410dc42d 100644
--- a/src/Namespace.js
+++ b/src/Namespace.js
@@ -56,7 +56,21 @@ export default class Namespace<Value> {
     }
 
     /**
-     * Get the current value of a name.
+     * Detect whether `name` has a definition.  Equivalent to
+     * `get(name) != null`.
+     */
+    has(name: string): boolean {
+        return this.current.hasOwnProperty(name) ||
+               this.builtins.hasOwnProperty(name);
+    }
+
+    /**
+     * Get the current value of a name, or `undefined` if there is no value.
+     *
+     * Note: Do not use `if (namespace.get(...))` to detect whether a macro
+     * is defined, as the definition may be the empty string which evaluates
+     * to `false` in JavaScript.  Use `if (namespace.get(...) != null)` or
+     * `if (namespace.has(...))`.
      */
     get(name: string): ?Value {
         if (this.current.hasOwnProperty(name)) {
diff --git a/src/environments/array.js b/src/environments/array.js
index 4be385dd..b0c959de 100644
--- a/src/environments/array.js
+++ b/src/environments/array.js
@@ -1,6 +1,7 @@
 // @flow
 import buildCommon from "../buildCommon";
 import defineEnvironment from "../defineEnvironment";
+import defineFunction from "../defineFunction";
 import mathMLTree from "../mathMLTree";
 import ParseError from "../ParseError";
 import ParseNode from "../ParseNode";
@@ -659,3 +660,18 @@ defineEnvironment({
     htmlBuilder,
     mathmlBuilder,
 });
+
+// Catch \hline outside array environment
+defineFunction({
+    type: "text", // Doesn't matter what this is.
+    names: ["\\hline", "\\hdashline"],
+    props: {
+        numArgs: 0,
+        allowedInText: true,
+        allowedInMath: true,
+    },
+    handler(context, args) {
+        throw new ParseError(
+            `${context.funcName} valid only within array environment`);
+    },
+});
diff --git a/src/macros.js b/src/macros.js
index 5c16c97c..6feac39c 100644
--- a/src/macros.js
+++ b/src/macros.js
@@ -38,6 +38,11 @@ export interface MacroContextInterface {
      */
     expandAfterFuture(): Token;
 
+    /**
+     * Recursively expand first token, then return first non-expandable token.
+     */
+    expandNextToken(): Token;
+
     /**
      * Fully expand the given macro name and return the resulting list of
      * tokens, or return `undefined` if no such macro is defined.
@@ -55,6 +60,14 @@ export interface MacroContextInterface {
      * and return the resulting array of arguments.
      */
     consumeArgs(numArgs: number): Token[][];
+
+    /**
+     * Determine whether a command is currently "defined" (has some
+     * functionality), meaning that it's a macro (in the current group),
+     * a function, a symbol, or one of the special commands listed in
+     * `implicitCommands`.
+     */
+    isDefined(name: string): boolean;
 }
 
 /** Macro tokens (in reverse order). */
@@ -118,11 +131,12 @@ defineMacro("\\TextOrMath", function(context) {
     }
 });
 
-// Basic support for global macro definitions:
-//     \gdef\macro{expansion}
-//     \gdef\macro#1{expansion}
-//     \gdef\macro#1#2{expansion}
-//     \gdef\macro#1#2#3#4#5#6#7#8#9{expansion}
+// Basic support for macro definitions:
+//     \def\macro{expansion}
+//     \def\macro#1{expansion}
+//     \def\macro#1#2{expansion}
+//     \def\macro#1#2#3#4#5#6#7#8#9{expansion}
+// Also the \gdef and \global\def equivalents
 const def = (context, global: boolean) => {
     let arg = context.consumeArgs(1)[0];
     if (arg.length !== 1) {
@@ -161,6 +175,7 @@ defineMacro("\\global", (context) => {
         throw new ParseError("Invalid command after \\global");
     }
     const command = next[0].text;
+    // TODO: Should expand command
     if (command === "\\def") {
         // \global\def is equivalent to \gdef
         return def(context, true);
@@ -169,6 +184,55 @@ defineMacro("\\global", (context) => {
     }
 });
 
+// \newcommand{\macro}[args]{definition}
+// \renewcommand{\macro}[args]{definition}
+// TODO: Optional arguments: \newcommand{\macro}[args][default]{definition}
+const newcommand = (context, existsOK: boolean, nonexistsOK: boolean) => {
+    let arg = context.consumeArgs(1)[0];
+    if (arg.length !== 1) {
+        throw new ParseError(
+            "\\newcommand's first argument must be a macro name");
+    }
+    const name = arg[0].text;
+
+    const exists = context.isDefined(name);
+    if (exists && !existsOK) {
+        throw new ParseError(`\\newcommand{${name}} attempting to redefine ` +
+            `${name}; use \\renewcommand`);
+    }
+    if (!exists && !nonexistsOK) {
+        throw new ParseError(`\\renewcommand{${name}} when command ${name} ` +
+            `does not yet exist; use \\newcommand`);
+    }
+
+    let numArgs = 0;
+    arg = context.consumeArgs(1)[0];
+    if (arg.length === 1 && arg[0].text === "[") {
+        let argText = '';
+        let token = context.expandNextToken();
+        while (token.text !== "]" && token.text !== "EOF") {
+            // TODO: Should properly expand arg, e.g., ignore {}s
+            argText += token.text;
+            token = context.expandNextToken();
+        }
+        if (!argText.match(/^\s*[0-9]+\s*$/)) {
+            throw new ParseError(`Invalid number of arguments: ${argText}`);
+        }
+        numArgs = parseInt(argText);
+        arg = context.consumeArgs(1)[0];
+    }
+
+    // Final arg is the expansion of the macro
+    context.macros.set(name, {
+        tokens: arg,
+        numArgs,
+    });
+    return '';
+};
+defineMacro("\\newcommand", (context) => newcommand(context, false, true));
+defineMacro("\\renewcommand", (context) => newcommand(context, true, false));
+defineMacro("\\providecommand", (context) => newcommand(context, true, true));
+
 //////////////////////////////////////////////////////////////////////
 // Grouping
 // \let\bgroup={ \let\egroup=}
diff --git a/test/helpers.js b/test/helpers.js
index 4f6d2f46..0300eb50 100644
--- a/test/helpers.js
+++ b/test/helpers.js
@@ -5,15 +5,10 @@ import ParseError from "../src/ParseError";
 import parseTree from "../src/parseTree";
 import Settings from "../src/Settings";
 
-export const defaultSettings = new Settings({
-    strict: false, // deal with warnings only when desired
-});
+export const nonstrictSettings = new Settings({strict: false});
 export const strictSettings = new Settings({strict: true});
 
-export const _getBuilt = function(expr, settings = defaultSettings) {
-    if (settings === defaultSettings) {
-        settings.macros = {};
-    }
+export const _getBuilt = function(expr, settings = new Settings()) {
     let rootNode = katex.__renderToDomTree(expr, settings);
 
     if (rootNode.classes.indexOf('katex-error') >= 0) {
@@ -43,7 +38,7 @@ export const _getBuilt = function(expr, settings = defaultSettings) {
  * @param settings
  * @returns {Object}
  */
-export const getBuilt = function(expr, settings = defaultSettings) {
+export const getBuilt = function(expr, settings = new Settings()) {
     expect(expr).toBuild(settings);
     return _getBuilt(expr, settings);
 };
@@ -54,7 +49,7 @@ export const getBuilt = function(expr, settings = defaultSettings) {
  * @param settings
  * @returns {Object}
  */
-export const getParsed = function(expr, settings = defaultSettings) {
+export const getParsed = function(expr, settings = new Settings()) {
     expect(expr).toParse(settings);
     return parseTree(expr, settings);
 };
@@ -73,7 +68,7 @@ export const stripPositions = function(expr) {
 };
 
 export const parseAndSetResult = function(expr, result,
-                                          settings = defaultSettings) {
+                                          settings = new Settings()) {
     try {
         return parseTree(expr, settings);
     } catch (e) {
@@ -89,7 +84,7 @@ export const parseAndSetResult = function(expr, result,
 };
 
 export const buildAndSetResult = function(expr, result,
-                                          settings = defaultSettings) {
+                                          settings = new Settings()) {
     try {
         return _getBuilt(expr, settings);
     } catch (e) {
diff --git a/test/katex-spec.js b/test/katex-spec.js
index c458ab2a..5ed1f342 100644
--- a/test/katex-spec.js
+++ b/test/katex-spec.js
@@ -11,7 +11,7 @@ import Options from "../src/Options";
 import Settings from "../src/Settings";
 import Style from "../src/Style";
 import {
-    defaultSettings, strictSettings,
+    strictSettings, nonstrictSettings,
     _getBuilt, getBuilt, getParsed, stripPositions,
 } from "./helpers";
 
@@ -1140,6 +1140,10 @@ describe("A begin/end parser", function() {
         expect("\\begin{matrix}\\hdashline a&b\\\\ \\hdashline c&d\\end{matrix}").toParse();
     });
 
+    it("should forbid hlines outside array environment", () => {
+        expect("\\hline").toNotParse();
+    });
+
     it("should error when name is mismatched", function() {
         expect("\\begin{matrix}a&b\\\\c&d\\end{pmatrix}").toNotParse();
     });
@@ -2282,7 +2286,7 @@ describe("A smash builder", function() {
 describe("A parser error", function() {
     it("should report the position of an error", function() {
         try {
-            parseTree("\\sqrt}", defaultSettings);
+            parseTree("\\sqrt}", new Settings());
         } catch (e) {
             expect(e.position).toEqual(5);
         }
@@ -2490,7 +2494,7 @@ describe("A macro expander", function() {
     const compareParseTree = function(actual, expected, macros) {
         const settings = new Settings({macros: macros});
         actual = stripPositions(parseTree(actual, settings));
-        expected = stripPositions(parseTree(expected, defaultSettings));
+        expected = stripPositions(parseTree(expected, new Settings()));
         expect(actual).toEqual(expected);
     };
 
@@ -2775,6 +2779,57 @@ describe("A macro expander", function() {
         expect(macros["\\foo"]).toBeFalsy();
     });
 
+    it("\\newcommand defines new macros", () => {
+        compareParseTree("\\newcommand\\foo{x^2}\\foo+\\foo", "x^2+x^2");
+        compareParseTree("\\newcommand{\\foo}{x^2}\\foo+\\foo", "x^2+x^2");
+        // Function detection
+        expect("\\newcommand\\bar{x^2}\\bar+\\bar").toNotParse();
+        expect("\\newcommand{\\bar}{x^2}\\bar+\\bar").toNotParse();
+        // Symbol detection
+        expect("\\newcommand\\lambda{x^2}\\lambda").toNotParse();
+        expect("\\newcommand\\textdollar{x^2}\\textdollar").toNotParse();
+        // Macro detection
+        expect("\\newcommand{\\foo}{1}\\foo\\newcommand{\\foo}{2}\\foo")
+            .toNotParse();
+        // Implicit detection
+        expect("\\newcommand\\limits{}").toNotParse();
+    });
+
+    it("\\renewcommand redefines macros", () => {
+        expect("\\renewcommand\\foo{x^2}\\foo+\\foo").toNotParse();
+        expect("\\renewcommand{\\foo}{x^2}\\foo+\\foo").toNotParse();
+        compareParseTree("\\renewcommand\\bar{x^2}\\bar+\\bar", "x^2+x^2");
+        compareParseTree("\\renewcommand{\\bar}{x^2}\\bar+\\bar", "x^2+x^2");
+        expect("\\newcommand{\\foo}{1}\\foo\\renewcommand{\\foo}{2}\\foo")
+            .toParseLike("12");
+    });
+
+    it("\\providecommand (re)defines macros", () => {
+        compareParseTree("\\providecommand\\foo{x^2}\\foo+\\foo", "x^2+x^2");
+        compareParseTree("\\providecommand{\\foo}{x^2}\\foo+\\foo", "x^2+x^2");
+        compareParseTree("\\providecommand\\bar{x^2}\\bar+\\bar", "x^2+x^2");
+        compareParseTree("\\providecommand{\\bar}{x^2}\\bar+\\bar", "x^2+x^2");
+        expect("\\newcommand{\\foo}{1}\\foo\\providecommand{\\foo}{2}\\foo")
+            .toParseLike("12");
+        expect("\\providecommand{\\foo}{1}\\foo\\renewcommand{\\foo}{2}\\foo")
+            .toParseLike("12");
+        expect("\\providecommand{\\foo}{1}\\foo\\providecommand{\\foo}{2}\\foo")
+            .toParseLike("12");
+    });
+
+    it("\\newcommand is local", () => {
+        expect("\\newcommand\\foo{1}\\foo{\\renewcommand\\foo{2}\\foo}\\foo")
+            .toParseLike("1{2}1");
+    });
+
+    it("\\newcommand accepts number of arguments", () => {
+        compareParseTree("\\newcommand\\foo[1]{#1^2}\\foo x+\\foo{y}",
+                         "x^2+y^2");
+        compareParseTree("\\newcommand\\foo[10]{#1^2}\\foo 0123456789", "0^2");
+        expect("\\newcommand\\foo[x]{}").toNotParse();
+        expect("\\newcommand\\foo[1.5]{}").toNotParse();
+    });
+
     // This may change in the future, if we support the extra features of
     // \hspace.
     it("should treat \\hspace, \\hskip like \\kern", function() {
@@ -2847,7 +2902,7 @@ describe("Unicode accents", function() {
             "\\tilde n" +
             "\\grave o\\acute o\\hat o\\tilde o\\ddot o" +
             "\\grave u\\acute u\\hat u\\ddot u" +
-            "\\acute y\\ddot y");
+            "\\acute y\\ddot y", nonstrictSettings);
     });
 
     it("should parse Latin-1 letters in text mode", function() {
@@ -2877,19 +2932,19 @@ describe("Unicode accents", function() {
     });
 
     it("should parse combining characters", function() {
-        expect("A\u0301C\u0301").toParseLike("Á\\acute C");
+        expect("A\u0301C\u0301").toParseLike("Á\\acute C", nonstrictSettings);
         expect("\\text{A\u0301C\u0301}").toParseLike("\\text{Á\\'C}", strictSettings);
     });
 
     it("should parse multi-accented characters", function() {
-        expect("ấā́ắ\\text{ấā́ắ}").toParse();
+        expect("ấā́ắ\\text{ấā́ắ}").toParse(nonstrictSettings);
         // Doesn't parse quite the same as
         // "\\text{\\'{\\^a}\\'{\\=a}\\'{\\u a}}" because of the ordgroups.
     });
 
     it("should parse accented i's and j's", function() {
-        expect("íȷ́").toParseLike("\\acute ı\\acute ȷ");
-        expect("ấā́ắ\\text{ấā́ắ}").toParse();
+        expect("íȷ́").toParseLike("\\acute ı\\acute ȷ", nonstrictSettings);
+        expect("ấā́ắ\\text{ấā́ắ}").toParse(nonstrictSettings);
     });
 });
 
@@ -3076,8 +3131,8 @@ describe("Symbols", function() {
 
 describe("strict setting", function() {
     it("should allow unicode text when not strict", () => {
-        expect("é").toParse(new Settings({strict: false}));
-        expect("試").toParse(new Settings({strict: false}));
+        expect("é").toParse(new Settings(nonstrictSettings));
+        expect("試").toParse(new Settings(nonstrictSettings));
         expect("é").toParse(new Settings({strict: "ignore"}));
         expect("試").toParse(new Settings({strict: "ignore"}));
         expect("é").toParse(new Settings({strict: () => false}));
@@ -3103,7 +3158,7 @@ describe("strict setting", function() {
     });
 
     it("should always allow unicode text in text mode", () => {
-        expect("\\text{é試}").toParse(new Settings({strict: false}));
+        expect("\\text{é試}").toParse(nonstrictSettings);
         expect("\\text{é試}").toParse(strictSettings);
         expect("\\text{é試}").toParse();
     });
diff --git a/test/mathml-spec.js b/test/mathml-spec.js
index a1f55196..cfd89ca2 100644
--- a/test/mathml-spec.js
+++ b/test/mathml-spec.js
@@ -8,13 +8,9 @@ import Options from "../src/Options";
 import Settings from "../src/Settings";
 import Style from "../src/Style";
 
-const defaultSettings = new Settings({});
-
-const getMathML = function(expr, settings) {
-    const usedSettings = settings ? settings : defaultSettings;
-
+const getMathML = function(expr, settings = new Settings()) {
     let startStyle = Style.TEXT;
-    if (usedSettings.displayMode) {
+    if (settings.displayMode) {
         startStyle = Style.DISPLAY;
     }
 
@@ -24,7 +20,7 @@ const getMathML = function(expr, settings) {
         maxSize: Infinity,
     });
 
-    const built = buildMathML(parseTree(expr, usedSettings), expr, options);
+    const built = buildMathML(parseTree(expr, settings), expr, options);
 
     // Strip off the surrounding <span>
     return built.children[0].toMarkup();
diff --git a/test/setup.js b/test/setup.js
index 731309ca..b79e429f 100644
--- a/test/setup.js
+++ b/test/setup.js
@@ -4,10 +4,10 @@
 import katex from "../katex";
 import ParseError from "../src/ParseError";
 import parseTree from "../src/parseTree";
+import Settings from "../src/Settings";
 import Warning from "./Warning";
 import stringify from 'json-stable-stringify';
 import {
-    defaultSettings,
     _getBuilt, buildAndSetResult, parseAndSetResult, stripPositions,
 } from "./helpers";
 
@@ -44,7 +44,7 @@ global.console.warn = jest.fn((warning) => {
 // Expect extensions
 
 expect.extend({
-    toParse: function(actual, settings = defaultSettings) {
+    toParse: function(actual, settings = new Settings()) {
         const result = {
             pass: true,
             message: () => `'${actual}' succeeded parsing`,
@@ -53,7 +53,7 @@ expect.extend({
         return result;
     },
 
-    toNotParse: function(actual, settings = defaultSettings) {
+    toNotParse: function(actual, settings = new Settings()) {
         const result = {
             pass: false,
             message: () =>
@@ -79,7 +79,7 @@ expect.extend({
     toFailWithParseError: function(actual, expected) {
         const prefix = "KaTeX parse error: ";
         try {
-            parseTree(actual, defaultSettings);
+            parseTree(actual, new Settings());
             return {
                 pass: false,
                 message: () => `'${actual}' parsed without error`,
@@ -115,7 +115,7 @@ expect.extend({
         }
     },
 
-    toBuild: function(actual, settings = defaultSettings) {
+    toBuild: function(actual, settings = new Settings()) {
         const result = {
             pass: true,
             message: () => `'${actual}' succeeded in building`,
@@ -124,7 +124,7 @@ expect.extend({
         return result;
     },
 
-    toNotBuild: function(actual, settings = defaultSettings) {
+    toNotBuild: function(actual, settings = new Settings()) {
         const result = {
             pass: false,
             message: () =>
@@ -147,7 +147,7 @@ expect.extend({
         return result;
     },
 
-    toParseLike: function(actual, expected, settings = defaultSettings) {
+    toParseLike: function(actual, expected, settings = new Settings()) {
         const result = {
             pass: true,
             message: () =>
@@ -174,7 +174,7 @@ expect.extend({
         return result;
     },
 
-    toBuildLike: function(actual, expected, settings = defaultSettings) {
+    toBuildLike: function(actual, expected, settings = new Settings()) {
         const result = {
             pass: true,
             message: () =>
@@ -201,7 +201,7 @@ expect.extend({
         return result;
     },
 
-    toWarn: function(actual, settings = defaultSettings) {
+    toWarn: function(actual, settings = new Settings()) {
         const result = {
             pass: false,
             message: () =>
diff --git a/test/unicode-spec.js b/test/unicode-spec.js
index 1453231e..5389d45a 100644
--- a/test/unicode-spec.js
+++ b/test/unicode-spec.js
@@ -4,7 +4,7 @@
 /* global describe: false */
 import Settings from "../src/Settings";
 import {scriptFromCodepoint, supportedCodepoint} from "../src/unicodeScripts";
-import {strictSettings} from "./helpers";
+import {strictSettings, nonstrictSettings} from "./helpers";
 
 describe("unicode", function() {
     it("should parse Latin-1 inside \\text{}", function() {
@@ -21,7 +21,7 @@ describe("unicode", function() {
 
     it("should parse Latin-1 outside \\text{}", function() {
         expect('ÀÁÂÃÄÅÈÉÊËÌÍÎÏÑÒÓÔÕÖÙÚÛÜÝàáâãäåèéêëìíîïñòóôõöùúûüýÿ' +
-            'ÇÐÞçðþ').toParse();
+            'ÇÐÞçðþ').toParse(nonstrictSettings);
     });
 
     it("should parse all lower case Greek letters", function() {