  In this paper we develop the convergence theory of simultaneous,
inhomogeneous Diophantine approximation on manifolds. A consequence of our main
result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension
strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy
condition, then $M$ is of Khintchine type for convergence. The key lies in
obtaining essentially the best possible upper bound regarding the distribution
of rational points near manifolds.
