This is a remarkable paper -- it solves a long-standing and celebrated open problem in the theory of billiard dynamical systems and mechanics. The authors prove that in a gas of \(N\) hard balls in the open space the number of possible collisions is uniformly bounded (until now, the problem had been only solved for \(N=3\)). The authors give an explicit upper bound for the number of collisions between \(N\) hard balls of arbitrary masses. They also solve a more general billiard problem: for multidimensional semidispersing billiards (i.e. with walls concave inward) the number of collisions near any ``nondegenerate'' corner point is uniformly bounded. A simple new criterion of nondegeneracy of a corner point is found. The authors give an elementary and very elegant solution of the above problems. In addition, they generalized the result (and the proof) to billiards on Riemannian manifolds with bounded sectional curvature, where the particle moves along geodesics between elastic collisions with walls. This involves the theory of Aleksandrov spaces.