  We consider the probability theory, and in particular the moment problem and
universality theorems, for random groups of the sort of that arise or are
conjectured to arise in number theory, and in related situations in topology
and combinatorics. The distributions of random groups that are discussed
include those conjectured in the Cohen-Lenstra-Martinet heuristics to be the
distributions of class groups of random number fields, as well as distributions
of non-abelian generalizations, and those conjectured to be the distributions
of Selmer groups of random elliptic curves. For these sorts of distributions on
finite and profinite groups, we survey what is known about the moment problem
and universality, give a few new results including new applications, and
suggest open problems.
