I will talk about some joint work with Jackson and Seward on some applications of descriptive set theory to problems in dynamical systems. We have previously shown that free Bernoulli subflows exist for all countably infinite groups. In this talk I will focus on two problems: one is to characterize the descriptive complexity of all (minimal) free Bernoulli subflows, and another is to determine the exact complexity of the isomorphism relation of (minimal) free Beroulli subflows. For the first problem the complete answer is given, which also introduces a new concept that seems to have never been studied in combinatorial group theory. For the second problem the complete answer is not known, but I will talk about partial results obtained.