For a finite extension of the rational numbers, its class group is a finite abelian group that plays an important role in the arithmetic of the field. We will discuss the question of the distribution of these groups as the field varies, and explain some conjectural answers to these questions, including the Cohen-Lenstra heuristics. We will define the moments of this distribution of groups, and explain the arithmetic meaning of the moments. We will discuss work bounding these moments, and briefly introduce the topics of the next three talks: questions of distribution of a non-abelian generalization of the class group and the resulting study of random non-abelian groups, theorems about these distributions for function fields over finite fields of growing size, and the probability theory of the Cohen-Lenstra distribution of finite abelian groups. The talk will include joint work with J. Ellenberg, L. Pierce, C. Turnage-Butterbaugh, Y. Liu, N. Boston, and H. Nguyen. The background assumed in the talk will be minimal (finite abelian groups, Galois theory, basic definitions in probability).