A stochastic process is a collection of random variables. We define and classify stochastic processes. Subsequently we provide extensive treatments of the following stochastic processes: the Poisson process, Markov chains, Markov processes, birth-and-death processes, random walks (classification of states, visits to a fixed state, transient and limiting behavior). We also discuss several applications of stochastic processes. The modeling of various every-day life situations (insurance, queues, biological phenomena) as stochastic process receives special attention.