A debris flow represents a mixture of sediment particles of various sizes and water flowing down a confined, channel-shaped region (e.g., gully, ravine or valley) down to its end, at which point it becomes unconfined and spreads out into a fan-shaped mass. This review begins with a survey of the literature on the physical-mathematical modeling of debris flows. Next, we discuss the basic aspects of their phenomenology, such as dilatancy, internal friction, fluidization, and particle segregation. The basic characterization of a debris flow as a mixture motivates the application of the continuum thermodynamical theory of mixtures to formulate a model for a debris flow as a viscous fluid-granular solid mixture. A major advantage of such a formulation, which goes beyond the most general models in the literature, e.g., Takahashi (1991), is that it can be used to expose and better understand the assumptions underlying existing models, as well as to derive new, more sophisticated models. Finally, we delve into the issue of how such models have been or can be implemented numerically, as well as general boundary conditions for debris flows.