Abstract: A common theme in mathematics is to try and find complete invariants for various classification problems (typically, one tries to understand isomorphism between some mathematical structures). One would then like to know whether the invariants used are as simple as possible; namely, one would like to know the precise complexity of the classification problem under consideration. There are many ways to address this type of question; I will try to motivate and explain the point of view of Borel complexity, focusing on classification problems induced by a group action, hoping to give an idea of the overall picture and of some open problems. The talk will be introductory and the speaker will aim at making it as self-contained as possible.