  In this article we study the deformation theory of conically singular Cayley
submanifolds. In particular, we prove a result on the expected dimension of a
moduli space of Cayley deformations of a conically singular Cayley submanifold.
Moreover, when the Cayley submanifold is a two-dimensional complex submanifold
of a Calabi--Yau four-fold we show by comparing Cayley and complex deformations
that in this special case the moduli space is a smooth manifold. We also
perform calculations of some of the quantities discussed for some examples.
