diff --git a/package.json b/package.json
index 0b2584b8298c1e..56b2e641ad59c9 100644
--- a/package.json
+++ b/package.json
@@ -83,6 +83,7 @@
     "confusing-browser-globals": "^1.0.9",
     "cross-env": "^6.0.0",
     "danger": "^9.1.8",
+    "dom-accessibility-api": "^0.2.0",
     "dtslint": "^2.0.0",
     "enzyme": "^3.9.0",
     "enzyme-adapter-react-16": "^1.14.0",
diff --git a/packages/material-ui-lab/src/Autocomplete/Autocomplete.test.js b/packages/material-ui-lab/src/Autocomplete/Autocomplete.test.js
index fee476af934dd8..84c6863e25fe01 100644
--- a/packages/material-ui-lab/src/Autocomplete/Autocomplete.test.js
+++ b/packages/material-ui-lab/src/Autocomplete/Autocomplete.test.js
@@ -120,9 +120,9 @@ describe('<Autocomplete />', () => {
 
       const buttons = getAllByRole('button');
       expect(buttons).to.have.length(2);
-      // TODO: computeAccessibleName
+      expect(buttons[0]).to.have.accessibleName('Clear');
       expect(buttons[0]).to.have.attribute('title', 'Clear');
-      // TODO: computeAccessibleName
+      expect(buttons[1]).to.have.accessibleName('Open');
       expect(buttons[1]).to.have.attribute('title', 'Open');
       buttons.forEach(button => {
         expect(button, 'button is not in tab order').to.have.property('tabIndex', -1);
@@ -161,9 +161,9 @@ describe('<Autocomplete />', () => {
 
       const buttons = getAllByRole('button');
       expect(buttons).to.have.length(2);
-      // TODO: computeAccessibleName
+      expect(buttons[0]).to.have.accessibleName('Clear');
       expect(buttons[0]).to.have.attribute('title', 'Clear');
-      // TODO: computeAccessibleName
+      expect(buttons[1]).to.have.accessibleName('Close');
       expect(buttons[1]).to.have.attribute('title', 'Close');
       buttons.forEach(button => {
         expect(button, 'button is not in tab order').to.have.property('tabIndex', -1);
diff --git a/packages/material-ui/src/Chip/Chip.test.js b/packages/material-ui/src/Chip/Chip.test.js
index 98600b09691a23..669213487dba1e 100644
--- a/packages/material-ui/src/Chip/Chip.test.js
+++ b/packages/material-ui/src/Chip/Chip.test.js
@@ -75,8 +75,7 @@ describe('<Chip />', () => {
 
       const button = getByRole('button');
       expect(button).to.have.property('tabIndex', 0);
-      // TODO: accessible name computation
-      expect(button).to.have.text('My Chip');
+      expect(button).to.have.accessibleName('My Chip');
     });
 
     it('should apply user value of tabIndex', () => {
diff --git a/packages/material-ui/src/Select/Select.test.js b/packages/material-ui/src/Select/Select.test.js
index a898521cb92a80..ad79a6137de0d5 100644
--- a/packages/material-ui/src/Select/Select.test.js
+++ b/packages/material-ui/src/Select/Select.test.js
@@ -406,6 +406,7 @@ describe('<Select />', () => {
     it('it will fallback to its content for the accessible name when it has no name', () => {
       const { getByRole } = render(<Select value="" />);
 
+      // TODO what is the accessible name actually?
       expect(getByRole('button')).to.have.attribute('aria-labelledby', ' ');
     });
 
diff --git a/packages/material-ui/src/TextField/TextField.test.js b/packages/material-ui/src/TextField/TextField.test.js
index fa47f18bacf7da..ddb2ea49d90989 100644
--- a/packages/material-ui/src/TextField/TextField.test.js
+++ b/packages/material-ui/src/TextField/TextField.test.js
@@ -160,14 +160,7 @@ describe('<TextField />', () => {
         </TextField>,
       );
 
-      const label = getByRole('button')
-        .getAttribute('aria-labelledby')
-        .split(' ')
-        .map(idref => document.getElementById(idref))
-        .reduce((partial, element) => `${partial} ${element.textContent}`, '');
-      // this whitespace is ok since actual AT will only use so called "flat strings"
-      // https://w3c.github.io/accname/#mapping_additional_nd_te
-      expect(label).to.equal(' Release:  Stable');
+      expect(getByRole('button')).to.have.accessibleName('Release: Stable');
     });
 
     it('creates an input[hidden] that has no accessible properties', () => {
diff --git a/packages/material-ui/test/integration/Select.test.js b/packages/material-ui/test/integration/Select.test.js
index c356ac6a3854a8..b8d4e52b8b5f3c 100644
--- a/packages/material-ui/test/integration/Select.test.js
+++ b/packages/material-ui/test/integration/Select.test.js
@@ -75,9 +75,7 @@ describe('<Select> integration', () => {
       const { getAllByRole, getByRole, queryByRole } = render(<SelectAndDialog />);
 
       const trigger = getByRole('button');
-      // basically this is a combined query getByRole('button', { name: 'Ten' })
-      // but we arent' there yet
-      expect(trigger).to.have.text('Ten');
+      expect(trigger).to.have.accessibleName('Ten');
       // Let's open the select component
       // in the browser user click also focuses
       fireEvent.mouseDown(trigger);
@@ -96,17 +94,31 @@ describe('<Select> integration', () => {
   });
 
   describe('with label', () => {
+    it('requires `id` and `labelId` for a proper accessible name', () => {
+      const { getByRole } = render(
+        <FormControl>
+          <InputLabel id="label">Age</InputLabel>
+          <Select id="input" labelId="label" value="10">
+            <MenuItem value="">none</MenuItem>
+            <MenuItem value="10">Ten</MenuItem>
+          </Select>
+        </FormControl>,
+      );
+
+      expect(getByRole('button')).to.have.accessibleName('Age Ten');
+    });
+
     // we're somewhat abusing "focus" here. What we're actually interested in is
     // displaying it as "active". WAI-ARIA authoring practices do not consider the
     // the trigger part of the widget while a native <select /> will outline the trigger
     // as well
     it('is displayed as focused while open', () => {
-      const { container, getByRole } = render(
+      const { getByTestId, getByRole } = render(
         <FormControl>
-          <InputLabel classes={{ focused: 'focused-label' }} htmlFor="age-simple">
+          <InputLabel classes={{ focused: 'focused-label' }} data-testid="label">
             Age
           </InputLabel>
-          <Select inputProps={{ id: 'age' }} value="">
+          <Select value="">
             <MenuItem value="">none</MenuItem>
             <MenuItem value={10}>Ten</MenuItem>
           </Select>
@@ -117,7 +129,7 @@ describe('<Select> integration', () => {
       trigger.focus();
       fireEvent.keyDown(document.activeElement, { key: 'Enter' });
 
-      expect(container.querySelector('[for="age-simple"]')).to.have.class('focused-label');
+      expect(getByTestId('label')).to.have.class('focused-label');
     });
 
     it('does not stays in an active state if an open action did not actually open', () => {
diff --git a/test/utils/createDOM.js b/test/utils/createDOM.js
index 4c7710bc93ab92..3b9a8cedf71aa7 100644
--- a/test/utils/createDOM.js
+++ b/test/utils/createDOM.js
@@ -2,7 +2,14 @@ const { JSDOM } = require('jsdom');
 const Node = require('jsdom/lib/jsdom/living/node-document-position');
 
 // We can use jsdom-global at some point if maintaining these lists is a burden.
-const whitelist = ['Element', 'HTMLElement', 'HTMLInputElement', 'Performance'];
+const whitelist = [
+  // required for fake getComputedStyle
+  'CSSStyleDeclaration',
+  'Element',
+  'HTMLElement',
+  'HTMLInputElement',
+  'Performance',
+];
 const blacklist = ['sessionStorage', 'localStorage'];
 
 function createDOM() {
diff --git a/test/utils/init.d.ts b/test/utils/init.d.ts
index 9db9ea9827ef5f..a83ccd15434d3e 100644
--- a/test/utils/init.d.ts
+++ b/test/utils/init.d.ts
@@ -2,6 +2,13 @@
 
 declare namespace Chai {
   interface Assertion {
+    /**
+     * checks if the accessible name computation (according to `accname` spec)
+     * matches the expectation.
+     * @see https://www.w3.org/TR/accname-1.2/
+     * @param name
+     */
+    accessibleName(name: string): Assertion;
     /**
      * checks if the element in question is considered aria-hidden
      * Does not replace accessibility check as that requires display/visibility/layout
diff --git a/test/utils/initMatchers.js b/test/utils/initMatchers.js
index 1ac2f5e8805a67..6490c97250d6f1 100644
--- a/test/utils/initMatchers.js
+++ b/test/utils/initMatchers.js
@@ -2,6 +2,7 @@ import chai from 'chai';
 import chaiDom from 'chai-dom';
 import { isInaccessible } from '@testing-library/dom';
 import { prettyDOM } from '@testing-library/react/pure';
+import { computeAccessibleName } from 'dom-accessibility-api';
 
 chai.use(chaiDom);
 chai.use((chaiAPI, utils) => {
@@ -62,4 +63,76 @@ chai.use((chaiAPI, utils) => {
       `expected ${utils.elToString(element)} to be accessible but it was inaccessible`,
     );
   });
+
+  chai.Assertion.addMethod('accessibleName', function hasAccessibleName(expectedName) {
+    const root = utils.flag(this, 'object');
+    // make sure it's an Element
+    new chai.Assertion(root.nodeType, `Expected an Element but got '${String(root)}'`).to.equal(1);
+
+    const blockElements = new Set(
+      'html',
+      'address',
+      'blockquote',
+      'body',
+      'dd',
+      'div',
+      'dl',
+      'dt',
+      'fieldset',
+      'form',
+      'frame',
+      'frameset',
+      'h1',
+      'h2',
+      'h3',
+      'h4',
+      'h5',
+      'h6',
+      'noframes',
+      'ol',
+      'p',
+      'ul',
+      'center',
+      'dir',
+      'hr',
+      'menu',
+      'pre',
+    );
+    /**
+     *
+     * @param {Element} element
+     * @returns {CSSStyleDeclaration}
+     */
+    function pretendVisibleGetComputedStyle(element) {
+      // `CSSStyleDeclaration` is not constructable
+      // https://stackoverflow.com/a/52732909/3406963
+      // this is not equivalent to the declaration from `getComputedStyle`
+      // e.g `getComputedStyle` would return a readonly declaration
+      // let's hope this doesn't get passed around until it's no longer clear where it comes from
+      const declaration = document.createElement('span').style;
+
+      // initial values
+      declaration.content = '';
+      // technically it's `inline`. We partially apply the default user agent sheet (chrome) here
+      // we're only interested in elements that use block
+      declaration.display = blockElements.has(element.tagName) ? 'block' : 'inline';
+      declaration.visibility = 'visible';
+
+      return declaration;
+    }
+
+    const actualName = computeAccessibleName(root, {
+      // in local development we pretend to be visible. full getComputedStyle is
+      // expensive and reserved for CI
+      getComputedStyle: process.env.CI ? undefined : pretendVisibleGetComputedStyle,
+    });
+
+    this.assert(
+      actualName === expectedName,
+      `expected ${utils.elToString(
+        root,
+      )} to have accessible name '${expectedName}' but got '${actualName}' instead.`,
+      `expected ${utils.elToString(root)} not to have accessible name '${expectedName}'.`,
+    );
+  });
 });
diff --git a/yarn.lock b/yarn.lock
index fd2a3e0b4b6f47..4953b33eff1e19 100644
--- a/yarn.lock
+++ b/yarn.lock
@@ -5751,6 +5751,11 @@ doctrine@^3.0.0:
   dependencies:
     esutils "^2.0.2"
 
+dom-accessibility-api@^0.2.0:
+  version "0.2.0"
+  resolved "https://registry.yarnpkg.com/dom-accessibility-api/-/dom-accessibility-api-0.2.0.tgz#2890ce677bd7b2172778ed979ab2ff4967c3085d"
+  integrity sha512-afrHGxXpS5C2jUC5hquPb3GWytNKHI+wJLKr/jvri95sZpLYpEJi3CtI/yBPEJ+/R9/CXaWXifadz94tsDcotg==
+
 dom-helpers@^3.2.1, dom-helpers@^3.4.0:
   version "3.4.0"
   resolved "https://registry.yarnpkg.com/dom-helpers/-/dom-helpers-3.4.0.tgz#e9b369700f959f62ecde5a6babde4bccd9169af8"