Quasiconformally invariant function spaces Pekka Koskela Abstract: It is almost immediate from the definition that the homogeneous Sobolev space consisting of functions with n-integrable gradients is invariant under quasiconformal mappings of an n-dimensional Euclidean space when n is at least two. This invariance actually characterizes quasiconformality. Reimann proved that also BMO is invariant. By interpolation, one concludes that also the diagonal Besov spaces with 0n/(n+s), but the case of Besov spaces was left open. I will explain this result and the even more recent result (Koch-Koskela-Saksman-Soto) for the Besov scales.