Advances in theory of evolution equations of many colliding particles

The review presents rigorous results of the theory of fundamental equations of evolution of many-particle systems with collisions and also considers their connection with nonlinear kinetic equations describing the collective behavior of particles in scaling approximations. This work is dedicated to the 160th anniversary of the birth of Dmytro Oleksandrovych Grave, the first academician of the Ukraine Academy of Sciences in mathematics and the founder of the Institute of Mathematics in 1920.