Maintain the aspect ratio of a div with CSS

I want to create a responsive div that can change its width/height as the window's width changes.

Are there any CSS rules that would allow the height to change according to the width, while maintaining its aspect ratio?

I know I can do this via JavaScript, but I would prefer using only CSS.

Just create a wrapper <div> with a percentage value for padding-bottom, like this:

.demoWrapper {
  padding: 10px;
  background: white;
  box-sizing: border-box;
  resize: horizontal;
  border: 1px dashed;
  overflow: auto;
  max-width: 100%;
  height: calc(100vh - 16px);
}

div {
  width: 100%;
  padding-bottom: 75%;
  background: gold; /** <-- For the demo **/
}

<div class="demoWrapper">
  <div></div>
</div>

It will result in a <div> with height equal to 75% of the width of its container (a 4:3 aspect ratio).

This relies on the fact that for padding :

The percentage is calculated with respect to the width of the generated box's containing block [...] (source: w3.org, emphasis mine)

Padding-bottom values for other aspect ratios and 100% width :

aspect ratio  | padding-bottom value
--------------|----------------------
    16:9      |       56.25%
    4:3       |       75%
    3:2       |       66.66%
    8:5       |       62.5%

Placing content in the div :

In order to keep the aspect ratio of the div and prevent its content from stretching it, you need to add an absolutely positioned child and stretch it to the edges of the wrapper with:

div.stretchy-wrapper {
  position: relative;
}

div.stretchy-wrapper > div {
  position: absolute;
  top: 0; bottom: 0; left: 0; right: 0;
}

Here's a demo and another more in depth demo