Darboux integrability for diagonal systems of hydrodynamic type

We prove that diagonal systems of hydrodynamic type are Darboux integrable if and only if the Laplace transformation sequences of the system for commuting flows terminate, give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, show that Darboux integrable systems are necessarily semihamiltonian, and discuss known and new examples.