  A map on a surface whose automorphism group has a subgroup acting regularly
on its vertices is called a Cayley map. Here we generalize that notion to
maniplexes and polytopes. We define $\mathcal{M}$ to be a \emph{Cayley
extension} of $\mathcal{K}$ if the facets of $\mathcal{M}$ are isomorphic to
$\mathcal{K}$ and if some subgroup of the automorphism group of $\mathcal{M}$
acts regularly on the facets of $\mathcal{M}$. We show that many natural
extensions in the literature on maniplexes and polytopes are in fact Cayley
extensions. We also describe several universal Cayley extensions. Finally, we
examine the automorphism group and symmetry type graph of Cayley extensions.
