vertical-align with Bootstrap 3

The below code worked for me:

.vertical-align {
    display: flex;
    align-items: center;
}

Update 2020

I know the original question was for Bootstrap 3, but now that Bootstrap 4 has been released, here is some updated guidance on vertical center.

Important! Vertical center is relative to the height of the parent

If the parent of the element your trying to center has no defined height, none of the vertical centering solutions will work!

Bootstrap 4

Now that Bootstrap 4 is flexbox by default there are many different approaches to vertical alignment using: auto-margins, flexbox utils, or the display utils along with vertical align utils. At first "vertical align utils" seems obvious, but these only work with inline and table display elements. Here are some Bootstrap 4 vertical centering options..

1 - Vertical Center Using Auto Margins:

Another way to vertically center is to use my-auto. This will center the element within its container. For example, h-100 makes the row full height, and my-auto will vertically center the col-sm-12 column.

<div class="row h-100">
    <div class="col-sm-12 my-auto">
        <div class="card card-block w-25">Card</div>
    </div>
</div>

Bootstrap 4 - Vertical center using auto-margins Demo

my-auto represents margins on the vertical y-axis and is equivalent to:

margin-top: auto;
margin-bottom: auto;

2 - Vertical Center with Flexbox:

Since Bootstrap 4 .row is now display:flex you can simply use align-self-center on any column to vertically center it...

       <div class="row">
           <div class="col-6 align-self-center">
                <div class="card card-block">
                 Center
                </div>
           </div>
           <div class="col-6">
                <div class="card card-inverse card-danger">
                    Taller
                </div>
          </div>
    </div>

or, use align-items-center on the entire .row to vertically center align all col-* in the row...

       <div class="row align-items-center">
           <div class="col-6">
                <div class="card card-block">
                 Center
                </div>
           </div>
           <div class="col-6">
                <div class="card card-inverse card-danger">
                    Taller
                </div>
          </div>
    </div>

Bootstrap 4 - Vertical center different height columns Demo

3 - Vertical Center Using Display Utils:

Bootstrap 4 has display utils that can be used for display:table, display:table-cell, display:inline, etc.. These can be used with the vertical alignment utils to align inline, inline-block or table cell elements.

<div class="row h-50">
    <div class="col-sm-12 h-100 d-table">
        <div class="card card-block d-table-cell align-middle">
            I am centered vertically
        </div>
    </div>
</div>

Bootstrap 4 - Vertical center using display utils Demo

Also see: Vertical Align Center in Bootstrap 4

Bootstrap 3

Flexbox method on the container of the item(s) to center:

.display-flex-center {
    display: flex;
    align-items: center;
}

Transform translateY method:

.transform-center-parent {
    position: relative;
    transform-style: preserve-3d;
}

.transform-center {
    position: relative;
    top: 50%;
    transform: translateY(-50%);
}

Display inline method:

.display-inline-block {
    display: inline;
}
.display-inline-block > div {
    display: inline-block;
    float: none;
    vertical-align: middle;
}

Demo of Bootstrap 3 centering methods

Try this in the CSS of the div:

display: table-cell;
vertical-align: middle;

I thought I'd share my "solution" in case it helps anyone else who isn't familiar with the @media queries themselves.

Thanks to @HashemQolami's answer, I built some media queries that would work mobile-up like the col-* classes so that I could stack the col-* for mobile but display them vertically-aligned in the center for larger screens, e.g.

<div class='row row-sm-flex-center'>
    <div class='col-xs-12 col-sm-6'></div>
    <div class='col-xs-12 col-sm-6'></div>
</div>

.

.row-xs-flex-center {
    display:flex;
    align-items:center;
}
@media ( min-width:768px ) {
    .row-sm-flex-center {
        display:flex;
        align-items:center;
    }
}
@media ( min-width: 992px ) {
    .row-md-flex-center {
        display:flex;
        align-items:center;
    }
}
@media ( min-width: 1200px ) {
    .row-lg-flex-center {
        display:flex;
        align-items:center;
    }
}

More complicated layouts that require a different number of columns per screen resolution (e.g. 2 rows for -xs, 3 for -sm, and 4 for -md, etc.) would need some more advanced finagling, but for a simple page with -xs stacked and -sm and larger in rows, this works fine.

Following the accepted answer, if you do not wish to customize the markup, for separation of concerns or simply because you use a CMS, the following solution works fine:

.valign {
  font-size: 0;
}

.valign > [class*="col"] {
  display: inline-block;
  float: none;
  font-size: 14px;
  font-size: 1rem;
  vertical-align: middle;
}

<link href="//maxcdn.bootstrapcdn.com/bootstrap/3.3.2/css/bootstrap.min.css" rel="stylesheet"/>
<div class="row valign">
   <div class="col-xs-5">
      <div style="height:5em;border:1px solid #000">Big</div>
   </div>
   <div class="col-xs-5">
       <div style="height:3em;border:1px solid #F00">Small</div>
   </div>
 </div>

The limitation here is that you cannot inherit font size from the parent element because the row sets the font size to 0 in order to remove white space.