^{Note: This is an updated answer. Comments below refer to an old version which messed around with keycodes.}

JavaScript

Try it yourself on JSFiddle.

You can filter the input values of a text <input> with the following setInputFilter function (supports Copy+Paste, Drag+Drop, keyboard shortcuts, context menu operations, non-typeable keys, the caret position, different keyboard layouts, and all browsers since IE 9):

// Restricts input for the given textbox to the given inputFilter function.
function setInputFilter(textbox, inputFilter) {
  ["input", "keydown", "keyup", "mousedown", "mouseup", "select", "contextmenu", "drop"].forEach(function(event) {
    textbox.addEventListener(event, function() {
      if (inputFilter(this.value)) {
        this.oldValue = this.value;
        this.oldSelectionStart = this.selectionStart;
        this.oldSelectionEnd = this.selectionEnd;
      } else if (this.hasOwnProperty("oldValue")) {
        this.value = this.oldValue;
        this.setSelectionRange(this.oldSelectionStart, this.oldSelectionEnd);
      } else {
        this.value = "";
      }
    });
  });
}

You can now use the setInputFilter function to install an input filter:

setInputFilter(document.getElementById("myTextBox"), function(value) {
  return /^\d*\.?\d*$/.test(value); // Allow digits and '.' only, using a RegExp
});

See the JSFiddle demo for more input filter examples. Also note that you still must do server side validation!

TypeScript

Here is a TypeScript version of this.

function setInputFilter(textbox: Element, inputFilter: (value: string) => boolean): void {
    ["input", "keydown", "keyup", "mousedown", "mouseup", "select", "contextmenu", "drop"].forEach(function(event) {
        textbox.addEventListener(event, function(this: (HTMLInputElement | HTMLTextAreaElement) & {oldValue: string; oldSelectionStart: number | null, oldSelectionEnd: number | null}) {
            if (inputFilter(this.value)) {
                this.oldValue = this.value;
                this.oldSelectionStart = this.selectionStart;
                this.oldSelectionEnd = this.selectionEnd;
            } else if (Object.prototype.hasOwnProperty.call(this, 'oldValue')) {
                this.value = this.oldValue;
                if (this.oldSelectionStart !== null &&
                    this.oldSelectionEnd !== null) {
                    this.setSelectionRange(this.oldSelectionStart, this.oldSelectionEnd);
                }
            } else {
                this.value = "";
            }
        });
    });
}

jQuery

There is also a jQuery version of this. See this answer.

HTML 5

HTML 5 has a native solution with <input type="number"> (see the specification), but note that browser support varies:

• Most browsers will only validate the input when submitting the form, and not when typing.
• Most mobile browsers don't support the step, min and max attributes.
• Chrome (version 71.0.3578.98) still allows the user to enter the characters e and E into the field. Also see this question.
• Firefox (version 64.0) and Edge (EdgeHTML version 17.17134) still allow the user to enter any text into the field.

Try it yourself on w3schools.com.

If you're running in a browser, then the easiest way is just to let the browser do it for you...

function stripHtml(html)
{
   let tmp = document.createElement("DIV");
   tmp.innerHTML = html;
   return tmp.textContent || tmp.innerText || "";
}

Note: as folks have noted in the comments, this is best avoided if you don't control the source of the HTML (for example, don't run this on anything that could've come from user input). For those scenarios, you can still let the browser do the work for you - see Saba's answer on using the now widely-available DOMParser.