The main objective in this thesis is to obtain a unified approach to the stability classification of continuous-time Markov chains defined on discrete state space via Foster-Lyapunov criteria. These criteria are tipically stated in terms of the generator of the process. Aleksandr Lyapunov first introduced these techniques for the study of ordinary differential equations and F. Gordon Foster first adapted them to a stochastic setting. After an introduction of the preliminaries about the main setting of work, a version of Dynkin's formula and its proof are provided. Ruling out unstable behaviours of the Markov chains such as explosivity or transience and establishing recurrence and positive recurrence is a complex task.