Plasters and renders and paints, copolymer dispersions and their use Compositions containing selected polymer dispersions, colloidal silica, fillers and pigments are described.These can be formulated to give plasters and renders or paints which are distinguished by excellent abrasion resistance, little tendency to soiling, high water vapor permeability, good adhesion and good weathering stability. The compositions according to the invention can be processed to give plasters and renders or paints which have a nanostructured surface and which differ from conventionally produced surfaces.