I describe recent research on "magnetomemory" materials: materials that combine ferromagnetism and shape-memory. The existence of this class of materials was evident from theoretical research in the early 1990s on the magnetoelastic behavior of the giant magnetostrictive material TbxDy1-xFe2 (x about 0.3). I shall describe the evidence, together with recent work on a "constrained theory of magnetostriction" that is guiding the search for new magnetomemory materials. This line of research involves, in a crucial way, modern concepts in the mathematical modeling of materials. Recently, an interesting example of a magnetomemory material has been discovered in the FePdCo system which exhibits, under small fields, magnetostrictive strains that are several times that of the best giant magnetostrictive materials. It appears possible by this strategy to find magnetostrictive materials that increase this factor to 100. Future directions of research are indicated: new systems of interest, thin film research, applications to micromachines. Joint work with Manfred Wuttig, Tom Shield, Rob Tickle, Pam Schumacher, Chris Palmstrøm and Hakim Bensaoula.