We present a review of recent progress in determining the
surface structure of quasicrystals, with emphasis on their
connections to mathematical tiling models. The review focusses
in particular on the five-fold surface of icosahedral
Al-Pd-Mn and the ten-fold surface of decagonal Al-Ni-Co. We
also assess their potential as templates for the formation of
two-dimensional quasicrystalline overlayers with reference to
recent investigations of atomic and molecular adsorption.