  In this paper we initiate a new approach to studying approximations by
rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main
result is a convergence Khintchine type theorem for arbitrary nondegenerate
submanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in the
theory of Diophantine approximation. Furthermore, we refine this result using
Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff
dimension of $\tau$-well approximable points lying on any nondegenerate
submanifold for a range of Diophantine exponents $\tau$ close to $1/n$. Our
approach uses geometric and dynamical ideas together with a new technique of
`generic and special parts'. In particular, we establish sharp upper bounds for
the number of rational points of bounded height lying near the generic part of
a non-degenerate manifold. In turn, we give an explicit exponentially small
bound for the measure of the special part of the manifold. The latter uses a
result of Bernik, Kleinbock and Margulis.
