This chapter discusses the foundation of stochastic processes. Much of statistical analysis is concerned with models in which the observations are assumed to vary independently. However, a great deal of data in economics, finance, engineering, and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest in itself. A model which describes the probability structure of a series of observations X t ,  t = 1, . . ., n,  is called a stochastic process. An X t might be the value of a stock price at time point t, the water level in a lake at time point t, and so on. The main purpose of this chapter is to provide a modern introduction to stochastic processes. Because statistical analysis for stochastic processes largely relies on asymptotic theory, we explain some useful limit theorems and central limit theorems.