The paper begins with an analysis of Riemann type problems for systems of hyperbolic conservation laws that arise when considering boundary conditions. These considerations lead naturally to an extension of Glimm's existence theorem to initial boundary value problems. The paper continues with results on the convergence asymptotically in time of the time average of the solution. In several sections at the end of the paper, the time average results are applied to various specific initial boundary value problems for scalar equations and systems, including the Buckley-Leverett equation, the polymer flood model, and the \(p\)-system of isentropic gas dynamics and one-dimensional nonlinear elasticity.