In this talk I will demonstrate the remarkable effectiveness of numerical continuation and boundary value formulations for computing stable and unstable manifolds. The first application concerns the so-called Lorenz manifold, for which the computations provide insight into the nature of the Lorenz attractor. The second application concerns the Circular Restricted Three-Body Problem (CR3BP), which models the motion of a satellite in an Earth-Moon-like system. Specifically we compute the unstable manifold of periodic orbits known as "Halo orbits", which have been used in actual space missions. The calculations lead to the detection of heteroclinic connections from a Halo orbit to invariant tori. Subsequent continuation of such connections (as the Halo orbit is allowed to change) leads to a variety of connecting orbits that may be of interest in space-mission design.