If it is one line of text and/or image, then it is easy to do. Just use:

text-align: center;
vertical-align: middle;
line-height: 90px;       /* The same as your div height */

That's it. If it can be multiple lines, then it is somewhat more complicated. But there are solutions on http://pmob.co.uk/. Look for "vertical align".

Since they tend to be hacks or adding complicated divs... I usually use a table with a single cell to do it... to make it as simple as possible.

Update for 2020:

Unless you need make it work on earlier browsers such as Internet Explorer 10, you can use flexbox. It is widely supported by all current major browsers. Basically, the container needs to be specified as a flex container, together with centering along its main and cross axis:

#container {
  display: flex;
  justify-content: center;
  align-items: center;
}

To specify a fixed width for the child, which is called a "flex item":

#content {
  flex: 0 0 120px;
}

Example: http://jsfiddle.net/2woqsef1/1/

To shrink-wrap the content, it is even simpler: just remove the flex: ... line from the flex item, and it is automatically shrink-wrapped.

Example: http://jsfiddle.net/2woqsef1/2/

The examples above have been tested on major browsers including MS Edge and Internet Explorer 11.

One technical note if you need to customize it: inside of the flex item, since this flex item is not a flex container itself, the old non-flexbox way of CSS works as expected. However, if you add an additional flex item to the current flex container, the two flex items will be horizontally placed. To make them vertically placed, add the flex-direction: column; to the flex container. This is how it works between a flex container and its immediate child elements.

There is an alternative method of doing the centering: by not specifying center for the distribution on the main and cross axis for the flex container, but instead specify margin: auto on the flex item to take up all extra space in all four directions, and the evenly distributed margins will make the flex item centered in all directions. This works except when there are multiple flex items. Also, this technique works on MS Edge but not on Internet Explorer 11.

Update for 2016 / 2017:

It can be more commonly done with transform, and it works well even in older browsers such as Internet Explorer 10 and Internet Explorer 11. It can support multiple lines of text:

position: relative;
top: 50%;
transform: translateY(-50%);

Example: https://jsfiddle.net/wb8u02kL/1/

To shrink-wrap the width:

The solution above used a fixed width for the content area. To use a shrink-wrapped width, use

position: relative;
float: left;
top: 50%;
left: 50%;
transform: translate(-50%, -50%);

Example: https://jsfiddle.net/wb8u02kL/2/

If the support for Internet Explorer 10 is needed, then flexbox won't work and the method above and the line-height method would work. Otherwise, flexbox would do the job.

Bootstrap 3

v3 Text Alignment Docs

<p class="text-left">Left aligned text.</p>
<p class="text-center">Center aligned text.</p>
<p class="text-right">Right aligned text.</p>
<p class="text-justify">Justified text.</p>
<p class="text-nowrap">No wrap text.</p>

Bootstrap 4

v4 Text Alignment Docs

<p class="text-xs-left">Left aligned text on all viewport sizes.</p>
<p class="text-xs-center">Center aligned text on all viewport sizes.</p>
<p class="text-xs-right">Right aligned text on all viewport sizes.</p>

<p class="text-sm-left">Left aligned text on viewports sized SM (small) or wider.</p>
<p class="text-md-left">Left aligned text on viewports sized MD (medium) or wider.</p>
<p class="text-lg-left">Left aligned text on viewports sized LG (large) or wider.</p>
<p class="text-xl-left">Left aligned text on viewports sized XL (extra-large) or wider.</p>

Bootstrap 5

v5 Text Alignment Docs

text-left has been replaced with text-start, text-right has been replaced with text-end

<p class="text-start">Left aligned text.</p>
<p class="text-center">Center aligned text.</p>
<p class="text-end">Right aligned text.</p>
<p class="text-justify">Justified text.</p>
<p class="text-nowrap">No wrap text.</p>