In this chapter, we discuss the Sturm–Liouville systems, which are composed of exactly one linear, second-order ordinary differential equation, together with prescribed boundary conditions given at the extremes of an interval of the real line, inside which we must look for the solution of that linear ordinary differential equation. These systems, also called Sturm–Liouville problems, are of vital importance to the discussion of Green’s functions. We also present the fundamental properties associated with their eigenvalues and eigenfunctions. The Green’s function for a linear ordinary differential equations is discussed and constructed. A brief mention to the construction of the generalized Green’s function is presented. Some solved exercises are discussed step by step, e.g. in the study of the Euler differential equations. As an application we obtain, by means of Green’s function, a relation involving an ordinary differential equation and an integral equation. We conclude with a list of proposed exercises; among them, some interesting applications involving the construction of the Green’s function are left to the reader, all of them with the answer and/or suggestion.