abstract: We consider norms of singular integral operators, quasiconvexity of variational integrals, a question of Morrey, and sharp regularity estimates of quasiconformal maps. All these different themes will be linked via Burkholder functionals. The talk is based on joint work with K. Astala (Helsinki), S. Geiss (Innsbruck), T. Iwaniec (Syracuse), S. Montgomery-Smith (Missouri) and I. Prause (Helsinki).