Sublattices and Subrings of Z^n and Random Finite Abelian Groups

How many sublattices of $\mathbb{Z}^n$ have index at most $X$?  If we choose such a lattice $\Lambda$ at random, what is the probability that $\mathbb{Z}^n/\Lambda$ is cyclic?  What is the probability that its order is odd?  Now let $R$ be a random subring of $\mathbb{Z}^n$.  What is the probability that $\mathbb{Z}^n/R$ is cyclic?  We will see how these questions fit into the study of random groups in number theory and combinatorics.  We will discuss connections to Cohen-Lenstra heuristics for class groups of number fields, sandpile groups of random graphs, and cokernels of random matrices over the integers.