  A homogeneous Riemannian manifold $(M=G/K, g)$ is called a space with
homogeneous geodesics or a $G$-g.o. space if every geodesic $\gamma (t)$ of $M$
is an orbit of a one-parameter subgroup of $G$, that is $\gamma(t) =
\exp(tX)\cdot o$, for some non zero vector $X$ in the Lie algebra of $G$. We
give an exposition on the subject, by presenting techniques that have been used
so far and a wide selection of previous and recent results.
  We also present some open problems.
