Not long after the discovery of quasicrystals, a specific orientation relationship between
crystalline and quasicrystalline structures was observed. The specific orientational
relationships were discovered in many systems, for instance surface alterations of bulk
quasicrystals, the growth of atomic overlayers on quasicrystalline substrates and
quasicrystalline thin films on crystalline substrates. In this paper, we review various
models described in the literature in explaining such quasicrystal–crystal epitaxy.