  In this paper we study the set of projective maps between compact proper
convex real projective manifolds. We show that this set contains only finitely
many distinct homotopy classes and each homotopy class has the structure of a
real projective manifold. When the target manifold is strictly convex, our
results imply that each non-trivial homotopy class contains at most one
projective map. These results are motivated by the theory of holomorphic maps
between compact complex manifolds.
