This book provides a systematic approach to the various methods available for deriving a Green's function. It begins by reviewing the historical development of the Green's function, the Fourier and Laplace transforms, the classical special functions of Bessel functions and Legendre polynomials, and the Dirac delta function. It then presents Green's functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain, including worked examples, problem sets, and illustrations.