  In this paper, we establish a sufficient condition for a geodesic in a
Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter
isometry group. As an application of this result, we provide a new proof of the
fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its
geodesics are homogeneous. We also study general properties of homogeneous
geodesics, in particular, the structure of the closure of a given homogeneous
geodesic. We present several examples where this closure is a torus of
dimension $\geq 2$ which is (respectively, is not) totally geodesic in the
ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups
supplied with left-invariant Riemannian metrics.
