This thesis develops the theory of Riemannian geometry on singular spaces (differentiable stacks) through the study of Riemannian groupoids and their Morita-invariant properties. With this aim, we consider generalized curves on Riemannian groupoids and define their normal length, relating it to the natural notion of distance in the orbit space. We introduce the notion of geodesic on Riemannian groupoids, verifying that it makes sense on the underlying Riemannian stack. We establish several foundational results, such as the existence and uniqueness of geodesics, a stacky Gauss Lemma, and a stacky Hopf-Rinow theorem. Using the stacky Hopf-Rinow theorem, we investigate the relations between invariant linearization for Lie groupoids and the existence of complete metrics.