Shape memory alloys and other active materials undergo displacive phase transitions which change the symmetry of the crystal lattice through a diffusionless coordinated motion of atoms. A signature feature of these materials is the hysteresis they exhibit in response to cyclic loading. The dissipation is due to propagating phase boundaries that can be represented at the continuum level as surfaces of discontinuity. Classical elastodynamics admits nonzero dissipation on moving discontinuities but provides no information about its origin and kinetics. In the presence of subsonic discontinuities, this results in an ill-posed initial-value problem. One can extract the missing information about phase boundary kinetics and regularize the continuum model by considering its natural discrete prototype. This leads to an incredibly rich lattice model that also describes other interesting phenomena such as phase nucleation and evolution of microstructures. In this talk I will describe some recent work in this direction.