How many sublattices of Zⁿ have index at most X?  If we choose such a lattice L at random, what is the probability that Zⁿ/L is cyclic?  What is the probability that its order is odd?  Now let R be a random subring of Zⁿ.  What is the probability that Zⁿ/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.