  We generalize classical large deviations theorems to the setting of complete
Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic
random walks via a general approach using visocity solutions for
Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of
Cram\'er's theorem. The approach also provides a new proof of Schilder's
theorem. Additionally, we provide a proof of Schilder's theorem by using an
embedding into Euclidean space, together with Freidlin-Wentzell theory.
