The Lorenz system is the classical example of a seemingly simple dynamical system that exhibits chaotic dynamics. In fact, there are numerous studies to characterize the complicated dynamics on the famous butterfly attractor. This talk addresses how the dynamics is organized more globally. An important role in this regard is played by the stable manifold of the origin, also known as the Lorenz manifold. In 1992 John Guckenheimer suggested this manifold as a bench-mark challenge for developing computational methods in dynamical systems. We show how the numerical continuation of orbit segments can be used to investigate and characterize the transition to chaos in the Lorenz system. Joint work with Eusebius Doedel (Concordia University, Montreal) and Bernd Krauskopf (University of Bristol).