Simulation of compressible two-phase and multimaterial flows have unique difficulties because each phase or material is governed by its own equation of state and the thermodynamical behavior is discontinuous at interfaces. The acceptance of shock-capturing numerical methods in gas dynamics has also benefited the compressible multi-phase flow community who developed augmented systems of governing equations to extend the shock-capturing strategy to multi-phase flows in a reliable manner. These extended systems, which I review here, are termed `diffuse interface' models, in the sense that they are designed to compute flow variables correctly in numerically diffused zones surrounding interfaces. In particular, they couple correctly the dynamics of the two fluids evolving on both sides of the (diffuse) interface and tend to the proper pure fluid governing equations far from said interfaces. This strategy has become efficient for contact interfaces separating fluids governed by different equations of state, in the presence or absence of capillary effects, and with phase change. More sophisticated materials than fluids (elastic-plastic materials for example) have been considered as well.