  Let $\cal C$ be a non--degenerate planar curve and for a real, positive
decreasing function $\psi$ let $\cal C(\psi)$ denote the set of simultaneously
$\psi$--approximable points lying on $\cal C$. We show that $\cal C$ is of
Khintchine type for divergence; i.e. if a certain sum diverges then the
one-dimensional Lebesgue measure on $\cal C$ of $\cal C(\psi)$ is full. We also
obtain the Hausdorff measure analogue of the divergent Khintchine type result.
In the case that $\cal C$ is a rational quadric the convergence counterparts of
the divergent results are also obtained. Furthermore, for functions $\psi$ with
lower order in a critical range we determine a general, exact formula for the
Hausdorff dimension of $\cal C(\psi)$. These results constitute the first
precise and general results in the theory of simultaneous Diophantine
approximation on manifolds.
