A complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems. The use of such a framework is well known for mechanical systems with inequality constraints, but we give a more general formulation which also applies, for instance, to switching control systems, The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state, A crucial technical tool is the so-called linear complementarity problem (LCP) from mathematical programming; we introduce various generalizations of this problem.