Quasiconformal (qc) maps are generalizations of conformal maps in the following sense: infinitesimal circles are mapped to infinitesimal ellipses in a bounded way. After a review on conformal maps, we start the talk by making sense of this. Through this notion, one can formulate the so-called measurable Riemann mapping theorem (MRMT), a helpful tool in surface topology, for instance. We'll discuss some heuristics to prove the MRMT. Finally, we will close with some applications of qc maps in geometry/topology. In spite of the MRMT, no background in measure theory or PDE will be assumed.