  We study hyperbolic systems of one-dimensional partial differential equations
under general, possibly non-local boundary conditions. A large class of
evolution equations, either on individual 1-dimensional intervals or on general
networks, can be reformulated in our rather flexible formalism, which
generalizes the classical technique of first-order reduction. We study forward
and backward well-posedness; furthermore, we provide necessary and sufficient
conditions on both the boundary conditions and the coefficients arising in the
first-order reduction for a given subset of the relevant ambient space to be
invariant under the flow that governs the system. Several examples are studied.
