  We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut
$\delta$-homogeneous spaces in the case of Riemannian manifolds. Every such
manifold has non-negative sectional curvature. The universal covering of any
$\delta$-homogeneous Riemannian manifolds is itself $\delta$-homogeneous. In
turn, every simply connected Riemannian $\delta$-homogeneous manifold is a
direct metric product of an Euclidean space and compact simply connected
indecomposable homogeneous manifolds; all factors in this product are itself
$\delta$-homogeneous. We find different characterizations of
$\delta$-homogeneous Riemannian spaces, which imply that any such space is
geodesic orbit (g.o.) and every normal homogeneous Riemannian manifold is
$\delta$-homogeneous. The g.o. property and the $\delta$-homogeneity property
are inherited by closed totally geodesic submanifolds. Then we find all
possible candidates for compact simply connected indecomposable Riemannian
$\delta$-homogeneous non-normal manifolds of positive Euler characteristic and
a priori inequalities for parameters of the corresponding family of Riemannian
$\delta$-homogeneous metrics on them (necessarily two-parametric). We prove
that there are only two families of possible candidates: non-normal
(generalized) flag manifolds $SO(2l+1)/U(l)$ and $Sp(l)/U(1)\cdot Sp(l-1)$,
$l\geq 2$, investigated earlier by W. Ziller, H. Tamaru, D.V. Alekseevsky and
A. Arvanitoyeorgos. At the end we prove that the corresponding two-parametric
family of Riemannian metrics on $SO(5)/U(2)=Sp(2)/U(1)\cdot Sp(1)$ satisfying
the above mentioned (strict!) inequalities, really generates
$\delta$-homogeneous spaces, which are not normal and are not naturally
reductive with respect to any isometry group.
