We investigate (N + 1)-dimensional anisotropic cosmological models with a massless scalar field, self-interacting through an exponential potential. The problem is reduced to an FRW model with an additional free, massless scalar field. In the case we find the exact general solution for the Robertson-Walker spacetime and the N>3 anisotropic Bianchi type I model which is a product of a flat (3 + 1)-dimensional manifold and an (N - 3)-dimensional torus. In both cases the solutions present singularities and power-law inflation. In the multidimensional anisotropic case we also analyse the conditions under which dimensional reduction can proceed. When N = 1 we consider the gravitational theory formed by setting the Ricci scalar equal to the trace of the energy-momentum tensor of the matter fields. In this case the exact general solution of the second-order system of gravitational and self-interacting scalar field equations exhibit singularities, their most notable departure from the case being the absence of both particle horizons and power-law inflationary solutions.