  For every $n\geq 3, g\geq 1$ and all large enough $e$ depending on $n,g$, a
general curve of genus $g$, degree $e$ in a hypersurface of degree $n$ in
$\mathbb P^n$, or in $\mathbb P^n$ itself, has semistable normal bundle. If $g,
e$ are large enough, the normal bundle is stable. Similar results hold for
curves on general hypersurfaces of degree $d< n$ in $\mathbb P^n$ provided
certain arithmetical conditions on $d, e$ hold. Previous results were
restricted to the case of ambient space $\mathbb P^n$ where either the normal
bundle has integer slope or $n=3$ or $g=1$ and $e=n+1$.
