This dissertation consists chiefly of three parts, which tell different facets in the development of one topic. The first part is an exploration of continuous dependence estimates of stochastically driven degenerate parabolic equations. The second is an extension of work done by Debussche and Vovelle on first order stochastic conservation laws - we extend their results to degenerate parabolic-hyperbolic conservation laws with additive noise, and derive results on the existence and uniqueness of invariant measures. In the third part we explore the long time behaviour of solutions to stochastic degenerate parabolic-hyperbolic conservation laws with multiplicative noise, depending non-linearly on the solution itself.