Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important implications...