Since 1980s work of Cohen-Lenstra and Friedman-Washington, many (pseudo-)random groups in number theory, combinatorics and topology have been conjectured---and sometimes proven---to match certain universal distributions, which appear as large-N limits of cokernels of N x N random matrices over $\mathbb{Z}$ or $\mathbb{Z}_p$. In this talk I discuss a new such distribution, the cokernel of a product of k independent matrices. For each fixed k, it converges to a universal distribution, generalizing in a natural way the k=1 case of the Cohen-Lenstra distribution. As time permits I will discuss the case when the number of products k goes to infinity along with N. Then the groups do not converge, but the fluctuations of their ranks and other statistics still approach limit distributions related to a new interacting particle system, the 'reflecting Poisson sea'. Based on https://arxiv.org/abs/2209.14957v2 (with Hoi Nguyen) and https://arxiv.org/abs/2312.11702, https://arxiv.org/abs/2310.12275.