A theoretical and experimental study is performed of rapid, fully developed flows of cohesionless granular materials down inclined chutes with a rough base. Two flow types are studied in detail: (1) immature sliding flow, where a stagnant zone forms on the base of the chute, and (2) fully developed sliding flow, where no such zone is formed. A simple phenomenological theory is developed that predicts the flow type and the associated velocity profile. The theory models dynamic stresses induced by interparticle collisions as well as quasi-static stresses induced by friction acting on semi-permanent interparticle contacts. Hence it is developed for the so-called frictional-collisional regime. Employing a photographical method, the flow type and the velocity profile are determined experimentally for various chute angles in a test set-up in which the granular material is continuously circulated. Quantitative agreement between the theoretical and the measured velocity profile is reasonably good, although it appears that at the largest chute angle the side wall friction causes deviations between the theoretical and the measured velocity profile.