Bootstrap 4:

<div class=" h-100 d-flex justify-content-center align-items-center">
  <div>
      Items are Centered horizontally and vertically
  </div>
</div>

JsFiddle

Bootstrap 5 (Updated 2021)

Bootstrap 5 is still flexbox based so vertical centering works the same way as it did in Bootstrap 4. For example, align-items-center (flex-direction: row) and justify-content-center (flex-direction: column) can used on the flexbox parent (row or d-flex).

Centering examples in Bootstrap 5

Vertical center (don't forget the parent must have a defined height!):

• my-auto for centering inside flex (.d-flex) elements
• my-auto can be used to center columns (.col-) inside row
• align-items-center to center columns (col-*) inside row

Horizontal center:

• text-center to center display:inline elements & column content
• mx-auto for centering inside flex elements
• mx-auto can be used to center columns (.col-) inside row
• justify-content-center to center columns (col-*) inside row

Bootstrap 4.3+ (Update 2019)

There's no need for extra CSS. What's already included in Bootstrap will work. Make sure the container(s) of the form are full height. Bootstrap 4 now has a h-100 class for 100% height...

Vertical center:

<div class="container h-100">
  <div class="row h-100 justify-content-center align-items-center">
    <form class="col-12">
      <div class="form-group">
        <label for="formGroupExampleInput">Example label</label>
        <input type="text" class="form-control" id="formGroupExampleInput" placeholder="Example input">
      </div>
      <div class="form-group">
        <label for="formGroupExampleInput2">Another label</label>
        <input type="text" class="form-control" id="formGroupExampleInput2" placeholder="Another input">
      </div>
    </form>   
  </div>
</div>

https://codeply.com/go/raCutAGHre

the height of the container with the item(s) to center should be 100% (or whatever the desired height is relative to the centered item)

Note: When using height:100% (percentage height) on any element, the element takes in the height of it's container. In modern browsers vh units height:100vh; can be used instead of % to get the desired height.

Therefore, you can set html, body {height: 100%}, or use the new min-vh-100 class on container instead of h-100.

Horizontal center:

• text-center to center display:inline elements & column content
• 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

Vertical Align Center in Bootstrap
Bootstrap 4 full-screen centered form
Bootstrap 4 center input group
Bootstrap 4 horizontal + vertical center full screen

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

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

Now, onto vertical centering...

Bootstrap 5 (Updated 2021)

Bootstrap 5 is still flexbox based so vertical centering works the same way as Bootstrap 4. For example, align-items-center, justify-content-center or auto margins can used on the flexbox parent (row or d-flex).

• use align-items-center on a flexbox row parent (row or d-flex)
• use justify-content-center on a flexbox column parent (d-flex flex-column)
• use my-auto on a flexbox parent

Vertical Center in Bootstrap 5

Bootstrap 4

You can use the new flexbox & size utilities to make the container full-height and display: flex. These options don't require extra CSS (except that the height of the container (ie:html,body) must be 100%).

Option 1 align-self-center on flexbox child

<div class="container d-flex h-100">
    <div class="row justify-content-center align-self-center">
     I'm vertically centered
    </div>
</div>

https://codeply.com/go/fFqaDe5Oey

Option 2 align-items-center on flexbox parent (.row is display:flex; flex-direction:row)

<div class="container h-100">
    <div class="row align-items-center h-100">
        <div class="col-6 mx-auto">
            <div class="jumbotron">
                I'm vertically centered
            </div>
        </div>
    </div>
</div>

https://codeply.com/go/BumdFnmLuk

Option 3 justify-content-center on flexbox parent (.card is display:flex;flex-direction:column)

<div class="container h-100">
    <div class="row align-items-center h-100">
        <div class="col-6 mx-auto">
            <div class="card h-100 border-primary justify-content-center">
                <div>
                    ...card content...
                </div>
            </div>
        </div>
    </div>
</div>

https://codeply.com/go/3gySSEe7nd

More on Bootstrap 4 Vertical Centering

Now that Bootstrap 4 offers flexbox and other utilities, there are many approaches to vertical alignment. http://www.codeply.com/go/WG15ZWC4lf

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

See this Q/A to center, but maintain equal height

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

More examples
Vertical center image in <div>
Vertical center .row in .container
Vertical center and bottom in <div>
Vertical center child inside parent
Vertical center full screen jumbotron

Important! Did I mention height?

Remember vertical centering is relative to the height of the parent element. If you want to center on the entire page, in most cases, this should be your CSS...

body,html {
  height: 100%;
}

Or use min-height: 100vh (min-vh-100 in Bootstrap 4.1+) on the parent/container. If you want to center a child element inside the parent. The parent must have a defined height.

Also see:
Vertical alignment in bootstrap 4
Bootstrap Center Vertical and Horizontal Alignment

If you're using Bootstrap 4 (it seems you are), you may use flex alignment classes like align-items-center justify-content-center

 <div class="col d-flex align-items-center justify-content-center">

More info: https://getbootstrap.com/docs/4.1/layout/grid/#alignment