Von Neumann asked how one could determine whether two measure-preserving transformations (of a standard Lebesgue space) are isomorphic. There are important classification results for transformations with pure point spectrum (Halmos-von Neumann [1942]) and for Bernoulli shifts (Kolmogorov [1958] and Ornstein [1970]). There are also important results showing that certain types of classifications are impossible for all (or even most) measure-preserving transformations. In this talk I'll survey some of this history and describe efforts (joint with Su Gao) to understand isomorphism on the class of rank-1 measure-preserving transformations. Almost every measure-preserving transformation is rank-1 and it was recently shown (non-constructively) that there is a Borel description of the isomorphism relation on this class.