There is much interest recently in the computation of invariant manifolds of vector fields. This thesis presents a continuation method for computing two-dimensional stable and unstable manifolds arising in dynamical systems.  The computations are illustrated using a well-known example, namely, the Lorenz system, for which detailed results are presented for the so-called "Lorenz manifold", i.e., the two-dimensional stable manifold of the origin, as well as for the two-dimensional unstable manifold of one of the two symmetric nonzero equilibria. A number of the infinitely many intersection curves of these manifolds are also determined accurately. All computations are carried out using the numerical continuation software AUTO, specifically its most recent version, AUTO-07p.  Various diagrams are given to illustrate the numerical results. Software based on OpenGL and Glut has been developed to visualize the numerically computed manifolds, using a triangulation of the data computed with AUTO.