Bernd Kirchheim Universität Leipzig Mathematisches Institut We investigate how Hausdorff dimension and measure behave if a set is mapped by a (exact representative of) map in a Sobolev space which embedds into the continuous functions (or equivalently L-infinity). The underlying results on decomposing the maps into pieces of appropriate Hoelder or Lipschitz continuity allow to establish essentially sharp versions of area formulae. This is joint work with G. Alberti, M. Csornyei and E. D'Aniello.