The theory of gravitational lensing is discussed in a Lorentzian manifold setting. To that end we fix a point ρ (observer at a particular instant) and a time- like curve γ (worldline of a light source) in a 4-dimensional Lorentzian manifold (spacetime) and we investigate how many past-pointing lightlike geodesics (light rays) go from ρ to γ. If there is more than one such geodesic, then we are in a gravitational lensing situation. Among other things, we study the geometry of light cones and we use the theory of conjugate points and cut points to find necessary and sufficient criteria for gravitational lensing; we discuss a Morse theory, based on a general relativistic version of Fermat’s principle, to characterize the number of images for gravitational lensing situations in globally hyperbolic spacetimes; and we discuss gravitational lensing in asymptotically simple and empty spacetimes, giving an elementary proof for an odd number theorem in this situation.