Let $G$ be a group, $X$ a set. Defining an action $G\times X \to X$ is the same as defining a group morphism $\rho: G\to Sym(X)$, through the formula $g\cdot - = \rho(g)$. The morphism $\rho$ is called a permutation representation.
Now, in what little representation theory I have studied one replaces the set $X$ with a vector space $V$, then defining a group morphism $\rho: G \to GL(V)$ is the same as defining an action $G\times V \to V$ such that $g\cdot-$ is linear for every g, through the same formula $g\cdot - = \rho(g)$. The morphism $\rho$ is called a linear representation.
The first question is: on what other objects can I make my group act? What are the conditions for the previous constructions to make sense when picking an object $X$ from a given category, and picking $Aut(X)$ the set of isomorphisms of $X$?
Addenda: 1) and when is this fruitful? On what categories have actions been studied?
2) Since we can always define through a morphism such a categorial representation, when is it equivalent to the definition of the form $G\times X \to X$?
For the second question: the construction of the semidirect product of groups $N$, $Q$ involves a morphism $Q \to Aut(N)$. One says that $Q$ acts on $N$ by automorphisms. I just noticed why one says so, and it's because it's just another case of an "action with extra structure". In this case it's the same giving such a morphism than giving an action $Q \times N \to N$ such that $q\cdot -$ is a group morphism for every $q$.
So the semidirect product is linked to representations in this way. The second question is, are there analogue constructions with other type of representations? What I mean is: making a group act on another group yields a new group (the semidirect product) which has as underlying set the direct product of both groups. Do actions over different categories yield new objects in a similar fashion?