The book is devoted to one of the current trends in modern theoretical physics - Poisson structures and their applications to various problems of Hamiltonian mechanics. These problems arise in solid-state dynamics, celestial mechanics, vortex theory, cosmological models. As a rule, the equations of motion of such systems can be written in a convenient polynomial (algebraic) form. This form is closely related to the possibility of representing equations of motion in the form of Hamilton equations with a linear Poisson structure associated with some Lie algebra. Nonlinear Poisson structures associated with infinite-dimensional Lie algebras are also discussed, the most typical cases of their occurrence are indicated. For the study of the obtained equations, the Penlev-Kovalevskaya method . is used. New cases of integrability of equations of dynamics and isomorphisms between various integrable problems are indicated. For specialists in mechanics and mathematics, dealing with the theory of dynamical systems, undergraduate and postgraduate universities .