In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In the first part of my talk, we discuss the quasiconformal equivalence for general open Riemann surfaces and give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. In the second part, we consider the quasiconformal equivalence of Riemann surfaces which are the complements of Cantor sets.