For everynumber field K, there is a finite abelian group C called the
class group, which serves as an obstruction to unique factorization.
Since Gauss, number theorists have tried to understand questions such
as how often is C trivial, or how often C contains an element of fixed
order (as K varies). In the 1970's, Cohen and Lenstra observed
empirically that when the degree and signature of K is fixed, the
isomorphism class of C adheres to a natural probability distribution.
I'll discuss these Cohen-Lenstra heuristics and survey what is known,
including recent attempts to use algebraic geometry as a way of
justifying and, perhaps eventually, proving them.