  On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb
P^n$ itself, we prove the existence of curves of any genus and high enough
degree depending on the genus passing through the expected number $t$ of
general points or incident to a general collection of subvarieties of suitable
codimensions. In some cases we also show that the family of curves through $t$
fixed points has general moduli as family of $t$-pointed curves. These results
imply positivity of certain intersection numbers on Kontsevich spaces of stable
maps. An arithmetical appendix by M. C. Chang descibes the set of numerical
characters ($n, d$, curve degree, genus) to which our results apply.
