This mini-course provides a presentation of the method of characteristics to
initial/boundary-value problems for systems of first-order partial differential equations
and to Hamilton-Jacobi variational inequalities. In particular, these results are indeed
useful for the treatment of hybrid systems of control theory. We rely on tools forged by
set-valued analysis and viability theory, which happen to be both efficient and versatile
to cover many problems. They find here unexpected relevance.