Have you ever tried to find a conformal mapping between a square and a non-square rectangle that maps corners to corners? This won't work, so the German mathematician Grötzsch asked for the most conformal way instead. That was in 1928, and he laid the foundation for what became later known as quasiconformal mappings, a natural generalization of conformal mappings. Some people love to use quasiconformal mappings as a tool for proving theorems, and (more often) other people love to compute them for applications in computer graphics. Both groups use it for the same reason; in some sense they are almost as good as conformal maps but much more flexible.
I will talk about the different approaches to quasiconformal mappings and try to get you interested in this topic.