Impulsive dynamical systems first appeared in the 1970's and had their mathematical foundation presented almost two decades later. Currently many authors are turning their attention to these systems, where a continuous evolution law is disrupted by abrupt changes of state, called \textit{jumps} or \textit{impulses}. \par In this paper the authors describe necessary and sufficient conditions under which one can achieve uniform stability (that is, solutions that start close to a given set remain close for all time, uniformly for the points in this set) and orbital stability (that is, given a set and a neighorbood $U$ of it, there is a smaller neighborhood $V\subset U$ such that all solutions that start in $V$ remain in $V$ for all times). \par They prove that relatively compact sets are uniformly stable iff they are orbitally stable, and in locally compact spaces they show that a relatively compact set is uniformly stable iff its prolongation coincides with its closure. \par Lastly, for a particular class of sets, the authors present conditions to ensure stability, orbital stability, and the existence of a Lyapunov function.