Progress on the development of a general framework for the simulation of turbulent, compressible, multi-phase, multi-material flows is described. It is based on interface-capturing and a compositional approach in which each component represents a different phase/fluid. It uses fully-unstructured meshes so that the latest mesh adaptivity methods can be exploited. A control volume-finite element mixed formulation is used to discretise the equations spatially. This employs finite-element pairs in which the velocity has a linear discontinuous variation and the pressure has a quadratic continuous variation. Interface-capturing is performed using a novel high-order accurate compressive advection method. Two-level time stepping is used for efficient time-integration, and a Petrov–Galerkin approach is used as an implicit large-eddy simulation model. Predictions of the numerical method are compared against experimental results for a five-material collapsing water column test case. Results from numerical simulations of two- and three-phase horizontal slug flows using this method are also reported and directions for future work are also outlined.