In this chapter, a characterization of sets of convergence of martingale is given in predictable terms. As a consequence, the strong LNL for square-integrable martingales is proved. This result is applied for derivation of strong consistency of the least-squares estimates in the framework of regression model with martingale errors. Moreover, the CLT for martingales is stated, and further this theorem together with the martingale LNL is applied to derive the asymptotic normality and strong consistency of martingale stochastic approximation procedures. A discrete version of the Girsanov theorem is given here with its further application for derivation of a discrete time Bachelier option pricing formula. In the last section, the notion of a martingale is extended in several directions: from asymptotic martingales and local martingales to martingale transforms and generalized martingales (see Borkar, Stochastic approximation: a dynamical systems viewpoint, 2008; Cohen and Elliott, Stochastic calculus and applications, 2nd edn., 2015; Edgar and Sucheston, J Multivariate Analy 6:193–221, 1976; Etheridge, A course in financial calculus, 2002; Jacod and Protter, Probability essentials, 2nd edn., 2003; Liptser and Shiryaev, Theory of martingales, 1989; Melnikov, Russian Math Surveys 51(5):43–136, 1996; Nevel’son and Has’minskii, Stochastic approximation and recursive estimation, 1976, and Shiryaev, Probability, 2nd edn., 1996).