Cayley submanifolds are naturally arising volume minimising submanifolds of $Spin(7)$- manifolds. In the special case that the ambient manifold is a four-dimensional Calabi--Yau manifold, a Cayley submanifold might be a complex surface, a special Lagrangian submanifold or neither. In this thesis, we study the deformation theory of Cayley submanifolds from two different perspectives.