Are (non-void) self-closing tags valid in HTML5?

The W3C validator (Wikipedia) shows a warning for self-closing tags (those that end with “/>”) on non-void elements. (Void elements are those that may not ever contain any content.) Are they still valid in HTML5?

Some examples of accepted void elements:

<br />
<img src="" />
<input type="text" name="username" />

Some examples of rejected non-void elements:

<div id="myDiv" />
<span id="mySpan" />
<textarea id="someTextMessage" />

_{Note:

The W3C validator actually accepts void self-closing tags: the author originally had a problem because of a simple typo (\> instead of />); however, self-closing tags are 100% not valid in HTML5 in general, and the answers elaborate on the issue of self-closing tags across various HTML flavors.}

Solution

(Theoretically) in HTML 4, <foo / (yes, with no > at all) means <foo> (which leads to <br /> meaning <br>> (i.e. <br>>) and <title/hello/ meaning <title>hello</title>). I use the term "theoretically" because this is an SGML rule that browsers did a very poor job of supporting. There was so little support (I only ever saw it work in emacs-w3m) that the spec advises authors to avoid the syntax.
In XHTML, <foo /> means <foo></foo>. This is an XML rule that applies to all XML documents. That said, XHTML is often served as text/html which (historically at least) gets processed by browsers using a different parser than documents served as application/xhtml+xml. The W3C provides compatibility guidelines to follow for XHTML as text/html. (Essentially: Only use self-closing tag syntax when the element is defined as EMPTY (and the end tag was forbidden in the HTML spec)).
In HTML5, the meaning of <foo /> depends on the type of element:
- On HTML elements that are designated as void elements (essentially "An element that existed before HTML5 and which was forbidden to have any content"), end tags are simply forbidden. The slash at the end of the start tag is allowed, but has no meaning. It is just syntactic sugar for people (and syntax highlighters) that are addicted to XML.
- On other HTML elements, the slash is an error, but error recovery will cause browsers to ignore it and treat the tag as a regular start tag. This will usually end up with a missing end tag causing subsequent elements to be children instead of siblings.
- Foreign elements (imported from XML applications such as SVG) treat it as self-closing syntax.