This paper presents a two-dimensional depth-integrated theory for the gravity-driven free-surface flow of a granular avalanche over an arbitrarily but gently curved and twisted topography which is an important extension of the original Savage & Hutter theory. In contrast to previous extensions the present coordinate system is based on a reference curve with curvature and torsion. Its derivation was necessary because real avalanches are often guided by rather curved and twisted valleys or more general slopes. The aim is to gain fundamental insight into the effects of non-uniform curvature and torsion, using an orthogonal coordinate system that rotates with torsion, and find an analytic description of flow avalanches. We present a set of model equations which comprises nonlinear hyperbolic partial differential equations for the space and time evolution of the granular pile height and the depth-averaged streamwise velocity distribution of a finite mass of granulates. The emerging theory is believed to be capable of predicting the flow of dense granular materials over moderately curved and twisted channels of general type.