The mechanisms of occurrence of nonlinear degenerate Poisson brackets in Hamiltonian mechanics, deformation of brackets and their cohomology. are considered. A geometric object is studied in detail, which is analogous to the Lie group for nonlinear brackets. It is presented with full evidence proposed by the authors of the construction of asymptotic quantization on common symplectic and poisson varieties, and in particular - the rule of quantization of two-dimensional films .. In addition, it contains an elementary introduction to the theory of quasiclassical approximation, a large reference material on the calculation of functions from non-commuting operators and a summary of the results on algebras with non-Liev permutation relations .. For specialists in differential geometry, algebra, mathematical physics, asymptotic methods; for graduate students and students of mathematical faculties