We discuss some recent results and current work on modeling PDEs with memory terms. These terms have frequently the form of convolution integral terms in which the unknowns and their derivatives model dependence of the underlying phenomena on the history of the process. Our focus is first on scalar conservation laws for which the memory terms describe the subscale diffusion present in highly heterogeneous media. We prove stability of numerical scheme for linear and nonlinear problems. Next we describe current work extending these results to systems as well as for a model describing flow and transport with non-separated scales.