  In this paper we study the geometry of the Severi varieties parametrizing
curves on the rational ruled surface $\fn$. We compute the number of such
curves through the appropriate number of fixed general points on $\fn$, and the
number of such curves which are irreducible. These numbers are known as Severi
degrees; they are the degrees of unions of components of the Hilbert scheme. As
(i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants
are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$
and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$
Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The
arguments are given in sufficient generality to also count plane curves in the
style of L. Caporaso and J. Harris and to lay the groundwork for computing
higher genus Gromov-Witten invariants of blow-ups of the plane at up to five
points (in a future paper).
