We present criteria for establishing a triangulation of a manifold. Given a manifold M , a simplicial complex A, and a map H from the underlying space of A to M , our criteria are presented in local coordinate charts for M , and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily checked algorithmically, they are expected to be of general use.