  In this article, we consider Cayley deformations of a compact complex surface
in a Calabi--Yau four-fold. We will study complex deformations of compact
complex submanifolds of Calabi--Yau manifolds with a view to explaining why
complex and Cayley deformations of a compact complex surface are the same. We
in fact prove that the moduli space of complex deformations of any compact
complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.
