  Extending results for space curves we establish bounds for the cohomology of
a non-degenerate curve in projective $n$-space. As a consequence, for any given
$n$ we determine all possible pairs $(d, g)$ where $d$ is the degree and $g$ is
the (arithmetic) genus of the curve. Furthermore, we show that curves attaining
our bounds always exist and describe properties of these extremal curves. In
particular, we determine the Hartshorne-Rao module, the generic initial ideal
and the graded Betti numbers of an extremal curve.
