In 1994, H. Ki and T. Linton proved that the set of numbers normal in base $b$ is a $\Pi_0^3$-complete set. We will overview some recent applications of descriptive set theory to the theory of normal numbers. In particular, we will discuss a strong generalization of the result of H. Ki and T. Linton to studying the complexity of sets of generic points for dynamical systems with a weak version of the specification property. As a special case, we are able to determine the Borel complexity of sets of normal numbers with respect to the regular continued fraction expansion, $\beta$-expansions, and generalized GLS expansions as well as solve an open problem of Sharkovsky-Sivak.
Joint with D. Airey, S. Jackson, and D. Kwietniak.