Scale image to fit a bounding box

Note: Even though this is the accepted answer, the answer below is more accurate and is currently supported in all browsers if you have the option of using a background image.

Edit 2: In the modern age, using object-fit might be an even better solution: https://developer.mozilla.org/en-US/docs/Web/CSS/object-fit

No, there is no CSS only way to do this in both directions. You could add

.fillwidth {
    min-width: 100%;
    height: auto;
}

To the an element to always have it 100% width and automatically scale the height to the aspect ratio, or the inverse:

.fillheight {
    min-height: 100%; 
    width: auto;
}

to always scale to max height and relative width. To do both, you will need to determine if the aspect ratio is higher or lower than it's container, and CSS can't do this.

The reason is that CSS does not know what the page looks like. It sets rules beforehand, but only after that it is that the elements get rendered and you know exactly what sizes and ratios you're dealing with. The only way to detect that is with JavaScript.

Although you're not looking for a JS solution I'll add one anyway if someone might need it. The easiest way to handle this with JavaScript is to add a class based on the difference in ratio. If the width-to-height ratio of the box is greater than that of the image, add the class "fillwidth", else add the class "fillheight".

$('div').each(function() {
  var fillClass = ($(this).height() > $(this).width()) 
    ? 'fillheight'
    : 'fillwidth';
  $(this).find('img').addClass(fillClass);
});

.fillwidth { 
  width: 100%; 
  height: auto; 
}
.fillheight { 
  height: 100%; 
  width: auto; 
}

div {
  border: 1px solid black;
  overflow: hidden;
}

.tower {
  width: 100px;
  height: 200px;
}

.trailer {
  width: 200px;
  height: 100px;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="tower">
  <img src="http://placekitten.com/150/150" />
</div>
<div class="trailer">
  <img src="http://placekitten.com/150/150" />
</div>

Here's a hackish solution I discovered:

#image {
    max-width: 10%;
    max-height: 10%;
    transform: scale(10);
}

This will enlarge the image tenfold, but restrict it to 10% of its final size - thus bounding it to the container.

Unlike the background-image solution, this will also work with <video> elements.

Interactive example:

 function step(timestamp) {
     var container = document.getElementById('container');
     timestamp /= 1000;
     container.style.left   = (200 + 100 * Math.sin(timestamp * 1.0)) + 'px';
     container.style.top    = (200 + 100 * Math.sin(timestamp * 1.1)) + 'px';
     container.style.width  = (500 + 500 * Math.sin(timestamp * 1.2)) + 'px';
     container.style.height = (500 + 500 * Math.sin(timestamp * 1.3)) + 'px';
     window.requestAnimationFrame(step);
 }

 window.requestAnimationFrame(step);

 #container {
     outline: 1px solid black;
     position: relative;
     background-color: red;
 }
 #image {
     display: block;
     max-width: 10%;
     max-height: 10%;
     transform-origin: 0 0;
     transform: scale(10);
 }

<div id="container">
    <img id="image" src="https://upload.wikimedia.org/wikipedia/en/7/7d/Lenna_%28test_image%29.png">
</div>

Thanks to CSS3 there is a solution !

The solution is to put the image as background-image and then set the background-size to contain.

HTML

<div class='bounding-box'>
</div>

CSS

.bounding-box {
  background-image: url(...);
  background-repeat: no-repeat;
  background-size: contain;
}

Test it here: http://www.w3schools.com/cssref/playit.asp?filename=playcss_background-size&preval=contain

Full compatibility with latest browsers: http://caniuse.com/background-img-opts

To align the div in the center, you can use this variation:

.bounding-box {
  background-image: url(...);
  background-size: contain;
  position: absolute;
  background-position: center;
  background-repeat: no-repeat;
  height: 100%;
  width: 100%;
}