In this paper we generalize some results of Richard Palais to the case of Lie supergroups and Lie superalgebras. More precisely, let $G$ be a Lie supergroup, $\mathfrak g$ its Lie superalgebra and let $ρ$ be an infinitesimal action (a representation) of $\mathfrak g$ on a supermanifold $M$. We will show that there always exists a local (smooth left) action of $G$ on $M$ such that $ρ$ is the map that associates the fundamental vector field on $M$ to an algebra element (we will say that the action integrates $ρ$). We also show that if $ρ$ is univalent, then there exists a unique maximal local action of $G$ on $M$ integrating $ρ$. And finally we show that if $G$ is simply connected and all (smooth, even) vector fields $ρ(X)$ are complete then there exists a global (smooth left) action of $G$ on $M$ integrating $ρ$. Omitting all references to the super setting will turn our proofs into variations of those of Palais.

56 pages, 14 figures (v1 only discusses global actions, v2 also discusses local actions)