Abstract: In this work we focus on analyzing the relationships between switching systems defined from a partition of the state space into convex cells, and relay or complementarity dynamical systems, which are other classes of discontinuous systems. First the conditions guaranteing the continuity of the vector field of the switching system at the cells boundaries (in which case the switching system is an ordinary differential equation with Lipschitz right-hand-side) are recalled. Then well-posedness results ({\em i.e.} results on the existence and the uniqueness of solutions) for different classes of relay and complementarity systems which are also switching systems are reviewed. The reverse issue (when can a switching system be rewritten equivalently as a relay or a complementarity system) is also tackled. Many examples from Mechanics, Circuits, Biology, illustrate the developments all through the paper. The paper focuses on systems with continuous solutions ({\em i.e.} with no state jumps). Convexity is the central property.