In this last chapter we present outlines of three important classical methods for the solution of partial differential equations (PDEs). We start with the method of separation of variables, which dates back to the 18th century and finds its principal application in the solution of second-order homogeneous and linear PDEs. A host of solutions to Fourier’s and Fick’s equations are arrived at by this method, and we present several illustrations to explain and expand on the procedure. An opening preamble is devoted to the twin topics of Fourier series and orthogonal functions, which play a key role in the application of the method of separation of variables.