  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 $\rho$ 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 $\rho$
is the map that associates the fundamental vector field on $M$ to an algebra
element (we will say that the action integrates $\rho$). We also show that if
$\rho$ is univalent, then there exists a unique maximal local action of $G$ on
$M$ integrating $\rho$. And finally we show that if $G$ is simply connected and
all (smooth, even) vector fields $\rho(X)$ are complete then there exists a
global (smooth left) action of $G$ on $M$ integrating $\rho$. Omitting all
references to the super setting will turn our proofs into variations of those
of Palais.
