  We consider nonlinear scalar conservation laws posed on a network. We
establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying
the entropy condition. We apply standard finite volume methods and show
stability and convergence to the unique entropy solution, thus establishing
existence of a solution in the process. Both our existence and
stability/uniqueness theory is centred around families of stationary states for
the equation. In one important case -- for monotone fluxes with an upwind
difference scheme -- we show that the set of (discrete) stationary solutions is
indeed sufficiently large to suit our general theory. We demonstrate the
method's properties through several numerical experiments.
