Naturally occurring fats are multi-component mixtures of triacylglycerols (TAGs), which are triesters of fatty acids with glycerol, and of which there are many chemically distinct compounds. Due to the importance of fats to the food and consumer products industries, fat crystallization has been studied for many years and many intricate features of TAG interactions, complicated by polymorphism, have been identified. The melting and crystallization properties of triacylglycerols are very sensitive to even small differences in fatty acid composition and position within the TAG molecule which cause steric hindrance. Differences of fatty acid chain length within a TAG lead to packing imperfections, and differences in chain lengths between different TAG molecules lead to a loss of intersolubility in the solid phase. The degree of saturation is hugely important as the presence of a double bond in a fatty acid chain causes rigid kinks in the fatty acid chains that produce huge disruption to packing structures with the result that TAGs containing double bonds have much lower melting points than completely saturated TAGs. All of these effects are more pronounced in the most stable polymorphic forms, which require the most efficient molecular packing. The crystallization of fats is complicated not just by polymorphism, but also because it usually occurs from a multi-component melt rather than from a solvent which is more common in other industrial crystallizations. This renders the conventional treatment of crystallization as a result of supersaturation somewhat meaningless. Most studies in the literature consequently quantify crystallization driving forces using the concept of supercooling below a distinct melting point. However whilst this is theoretically valid for a single component system, it can only at best represent a rough approximation for natural fat systems, which display a range of melting points. This paper reviews the latest attempts to describe the sometimes complex phase equilibria of fats using fundamental relationships for chemical potential that have so far been applied to individual species in melts of unary, binary and ternary systems. These can then be used to provide a framework for quantifying the true crystallization driving forces of individual components within a multi-component melt. These are directly related to nucleation and growth rates, and are also important in the prediction of polymorphic occurrence, crystal morphology and surface roughness. The methods currently used to evaluate induction time, nucleation rate and overall crystallization rate data are also briefly described. However, mechanistic explanations for much of the observed crystallization behaviour of TAG mixtures remain unresolved.