Make Bootstrap's Carousel both center AND responsive?

Now (on Boostrap 3 and 4) its simply :

.carousel-inner img {
  margin: auto;
}

I assume you have different sized images. I tested this myself, and it works as you describe (always centered, images widths appropriately)

/*CSS*/
div.c-wrapper{
    width: 80%; /* for example */
    margin: auto;
}

.carousel-inner > .item > img, 
.carousel-inner > .item > a > img{
width: 100%; /* use this, or not */
margin: auto;
}

<!--html-->
<div class="c-wrapper">
<div id="carousel-example-generic" class="carousel slide">
  <!-- Indicators -->
  <ol class="carousel-indicators">
    <li data-target="#carousel-example-generic" data-slide-to="0" class="active"></li>
    <li data-target="#carousel-example-generic" data-slide-to="1"></li>
    <li data-target="#carousel-example-generic" data-slide-to="2"></li>
  </ol>

  <!-- Wrapper for slides -->
  <div class="carousel-inner">
    <div class="item active">
      <img src="http://placehold.it/600x400">
      <div class="carousel-caption">
        hello
      </div>
    </div>
        <div class="item">
      <img src="http://placehold.it/500x400">
      <div class="carousel-caption">
        hello
      </div>
    </div>
    <div class="item">
      <img src="http://placehold.it/700x400">
      <div class="carousel-caption">
        hello
      </div>
    </div>
  </div>

  <!-- Controls -->
  <a class="left carousel-control" href="#carousel-example-generic" data-slide="prev">
    <span class="icon-prev"></span>
  </a>
  <a class="right carousel-control" href="#carousel-example-generic" data-slide="next">
    <span class="icon-next"></span>
  </a>
</div>
</div>

This creates a "jump" due to variable heights... to solve that, try something like this: Select the tallest image of a list

Or use media-query to set your own fixed height.

On Boostrap 4 simply add mx-auto to your carousel image class.

<div class="carousel-item">
  <img class="d-block mx-auto" src="http://placehold.it/600x400" />
</div> 

Combine with the samples from the bootstrap carousel documentation as desired.