Besov, or Lipschitz, spaces have been defined on closed subsets F of Rn, or on classes of closed subsets, in such a way that they are restrictions to F of Besov or Sobolev spaces on Rn. In this talk we discuss some spaces closely related to these Besov spaces, which have appeared in connection with analysis on fractals or wavelets on fractals, and give an overview of some applications of them to e.g. trace theorems.