  This manuscript is essentially a collection of lecture notes which were given
by the first author at the Summer School Wisl-2019, Poland and written down by
the second author. As the title suggests, the material covered here includes
the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors
and Poisson brackets), group actions and orbits (infinitesimal action,
stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian
actions. Finally, the phase space reduction is also discussed. The very last
section introduces the Poisson-Lie structures along with some related notions.
This text represents a brief review of a well-known material citing standard
references for more details. The exposition is concise, but pedagogical. The
Authors believe that it will be useful as an introductory exposition for
students interested in this specific topic.
