  Transitions between steady dynamical regimes in diverse applications are
often modelled using discontinuities, but doing so introduces problems of
uniqueness. No matter how quickly a transition occurs, its inner workings can
affect the dynamics of the system significantly. Here we discuss the way
transitions can be reduced to discontinuities without trivializing them, by
preserving so-called hidden terms. We review the fundamental methodology, its
motivations, and where their study seems to be heading. We derive a prototype
for piecewise smooth models from the asymptotics of systems with rapid
transitions, sharpening Filippov's convex combinations by encoding the tails of
asymptotic series into nonlinear dependence on a switching parameter. We
present a few examples that illustrate the impact of these on our standard
picture of smooth or only piecewise smooth dynamics.
