In this study, integral equation techniques are developed to solve two different types of groundwater flow problem. The first formulation, based on the Cauchy integral theorem, is used to obtain numerical solutions for unsteady, two-dimensional flow through a homogeneous, isotropic embankment. Free surface profiles following instantaneous drawdown of an upstream reservoir are presented in dimensionless form. The results from an unsteady, parallel-plate, viscous-flow experiment are included to check the accuracy of one of the numerical solutions and to show the limitations of the experimental approach. The numerical results are also compared with the simpler results obtained by making the Dupuit approximation, the inadequacies of which are also discussed. In aquifers where horizontal dimensions are orders of magnitude greater than vertical dimensions the Dupuit approximation is valid, and a three-dimensional problem is reduced to two dimensions in the horizontal plane. An integral equation technique for solving such problems is presented, in which time-dependent fundamental solutions of the governing partial differential equation are distributed around the flow boundary so that the initial and boundary conditions are satisfied. Two numerical methods of solving the integral equation are compared for a wide range of problems in both homogeneous and zoned regions. Each region may, or may not, contain wells. Comparisons with the finite-difference method are also made for certain problems.