This chapter is completely devoted to a discrete time stochastic analysis. It contains the key notions adapted to discrete time like stochastic basis, filtration, predictability, stopping times, martingales, sub- and supermartingales, local densities of probability measures, discrete stochastic integrals, and stochastic exponents. It is stated the Doob decomposition for stochastic sequences, maximal inequalities, and other Doob’s theorems. The developed martingale technique is further applied to prove several asymptotical properties for martingales and submartingales (see Baldi, An introduction through theory and exercises. Stochastic calculus, 2017; Çinlar, Probability and stochastics, vol. 261, 2011; Cohen and Elliott, Stochastic calculus and applications, 2nd edn., 2015; Durrett, Essentials of stochastic processes, 3rd edn., 2018; Jacod and Protter, Probability essentials, 2nd edn., 2003; Shiryaev, Probability, 2nd edn., 1996, and Williams, Probability and martingales, 1991).