The interplay between the topological structure and the algebraic structure of metrizable groups has been an active area of research for descriptive set theorists since the '90s. This study has produced a body of work in which many of the results have a classical (i.e., non-set theoretic) character, even if the proofs themselves utilize techniques and intuition from set theory. I will outline background notions in the field and survey some of my own work which is part of this program of research. My talk will be accessible and non-technical.