The class group of a number field, in colloquial terms, measures the failure of unique factorization in its ring of integers. Despite its central role in number theory, information about its structure, and even its size, mostly remain mysteries. In this talk, we will discuss conjectures on the asymptotics of the structure of class groups (the Cohen-Lenstra heuristics) as well as a more general notion of "categorical randomness" which has led to several other conjectures.