In the first part of thesis work we give an introduction to the theory of Markov chains with discrete state space. Further, we discuss their transience, recurrence, positive recurrence and null-recurrence property. The main result is that an irreducible Markov chain with finite state space or positive recurrent Markov chain with countable state space admits a unique stationary distribution and, in addition, by assuming aperiodicity of the chain, transition probabilities converge to the stationary distribution. In the second part of this thesis we recall the notion of martingales and discuss their basic properties. We state optional sampling theorem and discuss the relation between Markov chains and martingales. Further, we introduce a drift operator and discuss Foster-Lyapunov conditions for transience and recurrence. Drift condition for recurrence is based on the existence of nonnegative, unbounded function \(V\) and appropriate set \(C \in \mathcal{P}(S)\) so that \(E^x[V(X_1)]-V(x)\le0\), for every \(x \in C^c\), while drift condition for transience is based on the existence of nonnegative, bounded function \(V\) and the appropriate set \(C \in \mathcal{P}(S)\) so that \(E^x[V(X_1)]-V(x)\ge 0\), for every \(x \in C^c\). In the last part of the thesis, we analyze recurrence and transience of a random walk, random walk on the halfline and AR (1) process using Lyapunov function and Foster-Lyapunov conditions for recurrence and transience criteria.