Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems

A homogeneous Riemannian manifold $(M=G\mathrm{/}K,g)$ 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 $\gamma(t) = \exp(tX)\cdot o$,forsomenonzerovector$X$intheLiealgebraof$G$. 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.