Abstract: We review recent advances on the physical principles of crystallization in multicomponent systems, and use them to provide a framework for interpreting petrological and geochemical observations from igneous intrusions. The thermal structure of crystallizing boundary layers imposes strong constraints on the chemical and mineralogical compositions of the solid that can form from a given melt. The thermal problem is largely independent of the chemical composition of the melt, and sets the course of crystallization. A key problem to understand is the temperature of the solidification front (which we take to mean that point at which the last drop of liquid solidifies) particularly in the geologically relevant case in which the temperature at the cold boundary is below the eutectic temperature. Focussing on the solidification front rather than on the liquidus is a valuable perspective. Adcumulus growth requires specific conditions and much can be learned from trying to understand how these can develop from given starting conditions. We discuss the physical reasons and field evidence for the existence of mushy layers, where solid fraction and temperature vary by large amounts. In such regions of the magma chamber, thermodynamic equilibrium is nearly achieved locally and, for a given temperature, this specifies the composition of the interstitial melt. Thus, in a magma chamber, the whole liquid line of descent is present simultaneously. Compositional convection is likely to set in, and this exchange between the interior of the mushy layer and the main reservoir leads to a chemically stratified solid, and to adcumulus growth. The contribution of crystal settling to the floor cumulates is evaluated as a function of the magnitude of convective heat flux through the roof. It is shown that crystal settling is unlikely to overwhelm in-situ nucleation and growth at the floor.