  Geodesics become an essential element of the geometry of a semi-Riemannian
manifold. In fact, their differences and similarities with the (positive
definite) Riemannian case, constitute the first step to understand
semi-Riemannian Geometry. The progress in the last two decades has become
impressive, being especially relevant the systematic introduction of
(infinite-dimensional) variational methods.
  Our purpose is to give an overview, from refinements of classical results to
updated variational settings. First, several properties (and especially
completeness) of geodesics in some ambient spaces are studied. This includes
heuristic constructions of compact incomplete examples, geodesics in warped,
GRW or stationary spacetimes, properties in surfaces and spaceforms, or
problems on stability of completeness.
  Then, we study the variational framework, and focus on two fundamental
problems of this approach, which regards geodesic connectedness. The first one
deals with a variational principle for stationary manifolds, and its recent
implementation inside Causality Theory. The second one concerns orthogonal
splitting manifolds, and a reasonably self-contained development is provided,
collecting some steps spread in the literature.
