Abstract: For the past twenty years, a major focus of descriptive set theory has been the study of equivalence relations on Polish spaces that are definable (Borel, analytic, etc.) when viewed as sets of pairs; e.g. orbit equivalence relations induced by continuous actions of Polish groups are analytic. This study provides appropriate framework and tools for understanding the nature of classification of mathematical objects (measure-preserving transformations, unitary operators, Riemann surfaces, etc.) up to some notion of equivalence (isomorphism, conjugacy, conformal equivalence, etc.), and measuring the complexity of such classification problems. Due to its broad scope, it has natural interactions with other areas of mathematics, such as ergodic theory and topological dynamics, functional analysis and operator algebras. In this talk, I will give an introduction to this fascinating subject.