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Suppose that V_n is a sequence of representations of a family of groups such as the symmetric groups S_n, the general linear groups GL_n(Z), or the symplectic groups Sp_{2n}(Z). The field of representation stability centres around the questions 'what should it mean for the sequence {V_n} to stabilize?', and 'how can we detect this stability?' In this talk I will give some answers to these questions, and describe applications of recent progress to two families of groups, the Torelli groups of automorphisms of free groups, and the Torelli groups of mapping class groups of surfaces with one boundary component. This project uses a framework developed by Putman, Church-Ellenberg-Farb, and Putman-Sam. It is joint work with Jeremy Miller and Peter Patzt.