  A homogeneous Riemannian space $(M= G/H,g)$ is called a geodesic orbit space
(shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of
the isometry group $G$. We study the structure of compact GO-spaces and give
some sufficient conditions for existence and non-existence of an invariant
metric $g$ with homogeneous geodesics on a homogeneous space of a compact Lie
group $G$. We give a classification of compact simply connected GO-spaces $(M =
G/H,g)$ of positive Euler characteristic. If the group $G$ is simple and the
metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of
the flag manifolds $M_1=SO(2n+1)/U(n)$ or $M_2= Sp(n)/U(1)\cdot Sp(n-1)$ and
$g$ is any invariant metric on $M$ which depends on two real parameters. In
both cases, there exists unique (up to a scaling) symmetric metric $g_0$ such
that $(M,g_0)$ is the symmetric space $M = SO(2n+2)/U(n+1)$ or, respectively,
$\mathbb{C}P^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces.
