This chapter deals with inversion of the Laplace and Hankel transforms and their associated contour integration. The Laplace transform is a powerful method for solving a diffusion-like differential equation that contains a first order differential in time, for example, ∂h/∂t. We will introduce fundamentals of the Laplace transform, perform contour integration for the inverse Laplace transform, and describe a numerical technique for the inverse Laplace transform. Applications of the Laplace transform in equation solving are demonstrated in Chapters 4 and 5 on well hydraulics and Chapter 7 on solute transport.