  In the present review different effects related to the orbital degrees of
freedom are discussed. Leaving aside such aspects as the superexchange
mechanism of the cooperative Jahn-Teller distortions and different properties
of "Kugel-Khomskii"-like models, we mostly concentrate on other phenomena,
which are in the focus of the modern condensed matter physics. After a general
introduction we start with the discussion of the concept of effective reduction
of dimensionality due to orbital degrees of freedom and consider such phenomena
as the orbitally-driven Peierls effect and the formation of small clusters of
ions in the vicinity of a Mott transition, which behave like "molecules"
embedded in a solid. The second large section is devoted to the
orbital-selective effects such as the orbital-selective Mott transition and the
suppression of magnetism due to the fact that part of the orbitals start to
form singlet molecular orbitals. At the end the rapidly growing field of the
so-called "spin-orbit-dominated" transition metal compounds is briefly reviewed
including such topics as the interplay between the spin-orbit coupling and the
Jahn-Teller effect, the formation of the spin-orbit driven Mott and Peierls
states, the role of orbital degrees of freedom in generation of the Kitaev
exchange coupling, and the singlet (excitonic) magnetism in $4d$ and $5d$
transition metal compounds.
