Abstract: Random integer matrices have been studied since 1980s work by Cohen-Lenstra and Friedman-Washington as models for random groups in number theory and combinatorics, and present both similarities and differences with real/complex random matrices. In this talk I will give some background on the area and discuss a new universal distribution on collections of abelian groups which arises from random matrix products. Based on a joint work https://arxiv.org/abs/2209.14957v1 with Hoi Nguyen, and some earlier results of https://arxiv.org/abs/2011.09356, https://arxiv.org/abs/2112.02147.