Continuous casting is a mature, sophisticated technological process, used to produce most of the world's steel, so is worthy of fundamentally-based computational modeling. It involves many interacting phenomena including heat transfer, solidification, multiphase turbulent flow, clogging, electromagnetic effects, complex interfacial behavior, particle entrapment, thermal-mechanical distortion, stress, cracks, segregation, and microstructure formation. Furthermore, these phenomena are transient, three-dimensional, and operate over wide length and time scales. This paper reviews the current state of the art in modeling these phenomena, focusing on practical applications to the formation of defects. It emphasizes model verification and validation of model predictions. The models reviewed range from fast and simple for implementation into online model-based control systems to sophisticated multiphysics simulations that incorporate many coupled phenomena. Both the accomplishments and remaining challenges are discussed.