  Several physical problems such as the `twin paradox' in curved spacetimes
have purely geometrical nature and may be reduced to studying properties of
bundles of timelike geodesics. The paper is a general introduction to
systematic investigations of the geodesic structure of physically relevant
spacetimes. The investigations are focussed on the search of locally and
globally maximal timelike geodesics. The method of dealing with the local
problem is in a sense algorithmic and is based on the geodesic deviation
equation. Yet the search for globally maximal geodesics is non-algorithmic and
cannot be treated analytically by solving a differential equation. Here one
must apply a mixture of methods: spacetime symmetries (we have effectively
employed the spherical symmetry), the use of the comoving coordinates adapted
to the given congruence of timelike geodesics and the conjugate points on these
geodesics. All these methods have been effectively applied in both the local
and global problems in a number of simple and important spacetimes and their
outcomes have already been published in three papers. Our approach shows that
even in Schwarzschild spacetime (as well as in other static spherically
symetric ones) one can find a new unexpected geometrical feature: instead of
one there are three different infinite sets of conjugate points on each stable
circular timelike geodesic curve. Due to problems with solving differential
equations we are dealing solely with radial and circular geodesics.
