  Let $R$ be a compact, connected, orientable surface of genus $g$, $Mod_R^*$
be the extended mapping class group of $R$, $\mathcal{C}(R)$ be the complex of
curves on $R$, and $\mathcal{N}(R)$ be the complex of nonseparating curves on
$R$. We prove that if $g \geq 2$ and $R$ has at most $g-1$ boundary components,
then a simplicial map $\lambda: \mathcal{N}(R) \to \mathcal{N}(R)$ is
superinjective if and only if it is induced by a homeomorphism of $R$. We prove
that if $g \geq 2$ and $R$ is not a closed surface of genus two then
$Aut(\mathcal{N}(R))= Mod_R^*$, and if $R$ is a closed surface of genus two
then $Aut(\mathcal{N}(R))= Mod_R ^* /\mathcal{C}(Mod_R^*)$. We also prove that
if $g=2$ and $R$ has at most one boundary component, then a simplicial map
$\lambda: \mathcal{C}(R) \to \mathcal{C}(R)$ is superinjective if and only if
it is induced by a homeomorphism of $R$. As a corollary we prove some new
results about injective homomorphisms from finite index subgroups to $Mod_R^*$.
The last two results complete the author's previous results to connected
orientable surfaces of genus at least two.
