A mathematical model of the continuous casting process, which explicitly incorporates the presence of slag, molten steel, heat transfer through the mould walls, and shell solidification, is presented. The model is based on the solution of the Navier-Stokes equations for the multiphase slag-steel-air system under transient conditions, including tracking of the interface between these phases. The use of an extremely fine mesh (100 mm) in the meniscus region allows, for the first time, the direct calculation of liquid slag infiltration into the shell-mould gap. Elsewhere, a coarser mesh is used to capture the influence of the metal flow on the overall solution. Predictions are compared with prior, cold model experiments and high temperature mould simulators. Excellent agreement was found for features such as slag film development and heat flux variations during the oscillation cycle. Furthermore, predictions of shell thicknesses and heat fluxes for a variety of simulated casting speeds are also in good agreement with plant measurements. These findings provide an improved fundamental understanding of the basic principles involved in slag infiltration and solidification inside the mould and how these affect key process parameters, such as powder consumption and shell growth. These parameters have a decisive effect on the formation of oscillations marks and transverse cracks, which are a major source of defects in the casting practice.