This chapter presents a study of stability and controllability for semigroup actions on topological spaces. The notions of Poisson stable set, nonwandering set, and Lagrange stable set, which extend respectively the concepts of Poisson stable point, nonwandering point, and Lagrange stable motion are presented. These concepts of stability theory can be connected by means of limit sets. It is also presented an aspect of controllability by prolongations that is related to nonwandering points. The so-called prolongational control set is the link between Lyapunov stability and controllability.