Consider the three-dimensional flow of a viscous Newtonian fluid upon a curved two-dimensional substrate when the fluid film is thin, as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness $\eta$ and the average lateral velocity $\bar{\bm u}$. Centre manifold theory assures us that the model accurately and systematically includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model resolves wavelike phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, three-dimensional instabilities, Faraday waves, viscous hydraulic jumps, flow vortices in a compound channel and flow down and up a step. These models are the most complete models for thin-film flow of a Newtonian fluid; many other thin-film models can be obtained by different restrictions and truncations of the model derived here.