  The paper is devoted to the study of geodesic orbit Riemannian spaces that
could be characterize by the property that any geodesic is an orbit of a
1-parameter group of isometries. In particular, we discuss some important
totally geodesic submanifolds that inherit the property to be geodesic orbit.
For a given geodesic orbit Riemannian space, we describe the structure of the
nilradical and the radical of the Lie algebra of the isometry group. In the
final part, we discuss some new tools to study geodesic orbit Riemannian
spaces, related to compact Lie group representations with non-trivial principal
isotropy algebras. We discuss also some new examples of geodesic orbit
Riemannian spaces, new methods to obtain such examples, and some unsolved
questions.
