Given a transparent PNG displaying a simple shape in white, is it possible to somehow change the color of this through CSS? Some kind of overlay or what not?
You can use filters with -webkit-filter and filter:
Filters are relatively new to browsers but supported in over 90% of browsers according to the following CanIUse table: https://caniuse.com/#feat=css-filters
You can change an image to grayscale, sepia and lot more (look at the example).
So you can now change the color of a PNG file with filters.
body {
background-color:#03030a;
min-width: 800px;
min-height: 400px
}
img {
width:20%;
float:left;
margin:0;
}
/*Filter styles*/
.saturate { filter: saturate(3); }
.grayscale { filter: grayscale(100%); }
.contrast { filter: contrast(160%); }
.brightness { filter: brightness(0.25); }
.blur { filter: blur(3px); }
.invert { filter: invert(100%); }
.sepia { filter: sepia(100%); }
.huerotate { filter: hue-rotate(180deg); }
.rss.opacity { filter: opacity(50%); }

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