The theory of local polynomial approximation provides a unified framework for describing various spaces of smooth functions on Euclidean space. We use an analogous theory to define smoothness spaces on Ahlfors d-regular subsets of R n, n − 1 < d < n. This class of sets includes many interesting Cantor-type sets and self-similar sets. We give a characterization in terms of local polynomial approximations for traces of Besov spaces and Triebel–Lizorkin spaces on Ahlfors d-regular sets, n − 1 < d < n. We also focus on Sobolev-type spaces on such subsets and the relation between these spaces and traces of classical Sobolev spaces. This work extends the results of P. Shvartsman on characterizing traces of classical spaces of smooth functions on Ahlfors n-regular subsets of Rn. The talk is based on joint work with A.V. V¨ah¨akangas and on joint work with R. Korte.