Descriptive set theory offers tools of distinguishing naturally occurring isomorphism problems in mathematics. For example, the isomorphism problem for finitely generated groups has different complexity than the isomorphism problem for separable C*-algebras. During this talk I will try to make an overview of known results in this theory and explain why some isomorphism problems are “simpler” than others, and which problems are complete, i.e. the most difficult in some interesting complexity classes.