  Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two
distinct complex Grassmann manifolds of the same dimension where the target is
not the complex projective space. We show that, for any given $k,l$, the degree
of $f$ is zero provided that $m,n$ are sufficiently large. If the degree of $f$
is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also,
we prove that the image under $f^*$ of elements of a set of algebra generators
of $H^*(G_{m,l};\mathbb{Q})$ is determined upto a sign, $\pm$, if the degree of
$f$ is non-zero. Our proofs cover the case of quaternionic Grassmann manifolds
as well.
