The phenomenological theory proposed by Einstein for interpreting the phenomenon of Brownian motion is described in detail. The alternative approaches due to Langevin and Fokker–Planck are also illustrated. The theory of Markov chains is also reported as a basic mathematical approach to stochastic processes in discrete space and time; various of its applications, for example, the Monte Carlo method, are also illustrated. The theory of stochastic equations, as a representation of stochastic processes in continuous space–time, is discussed and used for obtaining a generalized, rigorous formulation of the Langevin and Fokker–Planck equations for generalized fluctuating observables. The Arrhenius formula as an example of the first exit-time problem is also derived.