  In this paper, we study holomorphic mappings sending a hyperquadric of
signature $\ell$ in $\bC^n$ into a hyperquadric of signature $\ell'$ in
$\bC^N$. We show (Theorem \ref{main}) that if the signature difference
$\ell'-\ell$ is not too large, then the mapping can be normalized by
automorphisms of the target hyperquadric to a particularly simple form and, in
particular, the image of the mapping is contained in a complex plane of a
dimension that depends only on $\ell$ and $\ell'$, and not on the target
dimension $N$. We also prove a Hopf Lemma type result (Theorem \ref{main2}) for
such mappings.
