A two–dimensional depth–integrated theory is derived for the gravity–driven free surface flow of cohesionless granular avalanches over complex shallow basal topography. This is an important extension of the one–dimensional Savage–Hutter theory. A simple curvilinear coordinate system is adopted, which is fitted to the ‘mean’ downslope chute topography. This defines a quasi–two–dimensional reference surface on top of which shallow three–dimensional basal topography is superposed. The governing equations are expressed in the curvilinear coordinate system and the mass– and momentum–balance equations are integrated through the avalanche depth. An ordering argument and a Mohr–Coulomb closure model are used to obtain a simple reduced system of equations. Laboratory experiments have been performed on a partly confined chute to validate the theory. An avalanche is released on a section inclined at 40 degrees to the horizontal, on which there is a concave parabolic cross–slope profile, and runs out through a smooth transition zone onto a horizontal plane. A comparison of the experiment with numerical solutions shows that the avalanche tail speed is under–predicted. A modification to the bed–friction angle is proposed, which brings theory and experiment into very good agreement. The partly confined chute channel the flow and results in significantly longer maximum run–out distances than on an unconfined chute. A simple shallow–water avalanche model is also derived and tested against the experimental results.