  We survey recent results on reaction-diffusion equations with discontinuous
hysteretic nonlinearities. We connect these equations with free boundary
problems and introduce a related notion of spatial transversality for initial
data and solutions. We assert that the equation with transverse initial data
possesses a unique solution, which remains transverse for some time, and also
describe its regularity. At a moment when the solution becomes nontransverse,
we discretize the spatial variable and analyze the resulting lattice dynamical
system with hysteresis. In particular, we discuss a new pattern formation
mechanism --- {\it rattling}, which indicates how one should reset the
continuous model to make it well posed.
