  The question whether a Riemannian manifold is geodesically connected can be
studied from geometrical as well as variational methods, and accurate results
can be obtained by using the associated distance and related properties of the
positive-definiteness. It is natural to state this problem in manifolds with
(possibly non-smooth) boundary, and some conditions on this boundary has been
studied.
  For Lorentzian manifolds, the results cannot be so general, and very
different techniques has been introduced which are satisfactory for some
classes of Lorentzian manifolds: spaceforms, disprisoning and pseudoconvex
manifolds, stationary, globally hyperbolic or multiwarped spacetimes... Some of
them are appliable to semi-Riemannian manifolds with higher index or even to
manifolds with just an affine connection.
  Our purpose is to review these semi-Riemannian techniques, discussing the
results and possible extensions.
