Bootstrap 5 (update 2021)
Bootstrap 5 still uses flexbox grid layout, so centering works the same way.
text-center to center display:inline elements (ie: span, img)
mx-auto for centering inside flex elements
offset-* or mx-auto can be used to center columns (.col-)
justify-content-center to center columns (col-*) inside row
Bootstrap 4
"Centered content" can mean many different things, and Bootstrap centering has changed a lot since the original post.
Horizontal Center
Bootstrap 3
text-center is used for display:inline elements
center-block to center display:block elements
col-*offset-* to center grid columns
- see this answer to center the navbar
Demo Bootstrap 3 Horizontal Centering
Bootstrap 4
text-center is still used for display:inline elements
mx-auto replaces center-block to center display:block elements
offset-* or mx-auto can be used to center grid columns
justify-content-center in row can also be used to center col-*
mx-auto (auto x-axis margins) will center display:block or display:flex elements that have a defined width, (%, vw, px, etc..). Flexbox is used by default on grid columns, so there are also various flexbox centering methods.
Demo Bootstrap 4 Horizontal Centering
**Vertical Center**
--
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 it's 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>
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>
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>
Vertical Center Using Display Utils Demo