  In this paper, we study homogeneous geodesics in homogeneous Finsler spaces.
We first give a simple criterion that characterizes geodesic vectors. We show
that the geodesics on a Lie group, relative to a bi-invariant Finsler metric,
are the cosets of the one-parameter subgroups. The existence of infinitely many
homogeneous geodesics on compact semi-simple Lie group is established. We
introduce the notion of naturally reductive homogeneous Finsler space. As a
special case, we study homogeneous geodesics in homogeneous Randers spaces.
Finally, we study some curvature properties of homogeneous geodesics. In
particular, we prove that the S-curvature vanishes along the homogeneous
geodesics.
