  In a general-relativistic spacetime (Lorentzian manifold), gravitational
lensing can be characterized by a lens map, in analogy to the lens map of the
quasi-Newtonian approximation formalism. The lens map is defined on the
celestial sphere of the observer (or on part of it) and it takes values in a
two-dimensional manifold representing a two-parameter family of worldlines. In
this article we use methods from differential topology to characterize global
properties of the lens map. Among other things, we use the mapping degree (also
known as Brouwer degree) of the lens map as a tool for characterizing the
number of images in gravitational lensing situations. Finally, we illustrate
the general results with gravitational lensing (a) by a static string, (b) by a
spherically symmetric body, (c) in asymptotically simple and empty spacetimes,
and (d) in weakly perturbed Robertson-Walker spacetimes.
