A $G$-category is a category on which a group $G$ acts. This work studies the $2$-category $G$-Cat of $G$-categories, $G$-functors (functors which commute with the action of $G$) and $G$-natural transformations (natural transformations which commute with the $G$-action). There is particular emphasis on the relationship between a $G$-category and its stable subcategory, the largest sub-$G$-category on which $G$ operates trivially. Also contained here are some very general applications of the theory to various additive $G$-categories and to $G$-topoi.
$G$-Categories: The stable subcategory, $G$-limits and stable limits Systems of isomorphisms and stably closed $G$-categories Partial $G$-sets: $G$-adjoints and $G$-equivalence Par$(G$-set) and $G$-representability Transversals Transverse limits and representations of transversaled functors Reflections and stable reflections $G$-Cotripleability The standard factorization of insertion Cotripleability of stable reflectors The case of $\scr D^G$ Induced stable reflections and their signatures The $\scr D^G$-targeted case.