The SageManifolds project aims at extending the modern computer algebra system SageMath towards differential geometry and tensor calculus. All SageManifolds code is included in SageMath, i.e. it does not require any separate installation. SageManifolds deals with differentiable manifolds of arbitrary dimension. Various coordinate charts and vector frames can be introduced on the manifold, which does not need to be parallelizable. A given tensor field is then described by its sets of components in each vector frame, with automatic change-of-frame transformations for overlapping vector frames. Generic pseudo-Riemannian manifolds can be considered, among which Riemannian manifolds and Lorentzian manifolds, with applications to General Relativity. In particular, the computation of the Riemann curvature tensor and associated tensors (Ricci, Weyl, Schouten and Cotton tensors) is implemented. SageManifolds can also deal with generic affine connections, not necessarily Levi-Civita ones.