This paper considers the approximate reconstruction of points, x \in R^D, which are close to a given compact d-dimensional submanifold, M, of R^D using a small number of linear measurements of x. In particular, it is shown that a number of measurements of x which is independent of the extrinsic dimension D suffices for highly accurate reconstruction of a given x with high probability. Furthermore, it is also proven that all vectors, x, which are sufficiently close to M can be reconstructed with uniform approximation guarantees when the number of linear measurements of x depends logarithmically on D. Finally, the proofs of these facts are constructive: A practical algorithm for manifold-based signal recovery is presented in the process of proving the two main results mentioned above.