  Actions of monoidal categories on categories, also known as actegories, have
been familiar to category theorists for a long time, and yet a comprehensive
overview of this topic seems to be missing from the literature. Recently,
actegories have been increasingly employed in applied category theory, thereby
encouraging an effort to fill this gap according to the new needs of these
applications. This work started as an investigation of the notion of monoidal
actegory, a compatible pair of monoidal and actegorical structures, and ended
up including a sizable reference on the elementary theory of actegories. We
cover basic definitions and results on actegories and biactegories, spelling
out explicitly many folkloric definitions, including their tensor product and
their hom-tensor adjunction. We give new definitions of actegories with
monoidal, braided monoidal and symmetric monoidal structure. In the last
section, we provide three Cayley-like classification results for these
structures.
