In this chapter we give an introduction to the area of matrix functions. We first define general matrix functions and derive their most important properties. Using the examples of network analysis and chemical reactions, we illustrate how matrix functions arise naturally in applications. The network analysis example involves the exponential function of matrices, and we study the properties of this important function in detail. The analysis of chemical reaction kinetics leads to a system of ordinary differential equations, whose solution again is based on the matrix exponential function.