A short sketch of main notions of an equilibrium statistical mechanics, which are connected with the construction of Gibbs measure in the fixed volume and on the space of infinite configurations, is given. A short overview of works of predecessors concerning the criteria of stability, superstabilty and strong superstability is also proposed. Connection between the conditions of stability, superstabilty and strong superstability of pair interaction and the existence of at least one Gibbs state has been analyzed. There has been established a close connection between the research of sufficient conditions of stability of interaction of infinite systems and a classical theory of minimizing measures on compact sets. It is shown in the following way: the partition of the space into non-intersecting hypercubes is introduced; then the energy of a fixed configuration of particles can be divided into two parts, one of them is the interaction between the particles inside each separate cube, and another one is the interaction between the particles from different cubes. The first sum can be estimated from below by a minimal energy of a fixed number of particles in the hypercube. A complete survey of the previous results concerning this subject in the classical and modern potential theory is given. New sufficient conditions of stability, superstability and strong superstability of interaction, which is defined by pair potential, are obtained. The main advantage of these criteria in comparison with the previous results in Dobrushin's and Ruelle's works is that the new exact values of the constants and their dependencies on the parameters, which define the interaction potential, are proposed. It was shown, that if the pair potential is of Lenard-Jones type (on small distances this potential is of Riesz type and it satisfies the integrability condition) then the constant in the condition of stability does not depend on the value of parameter a of the partition of the space into hypercubes. In the case of many-body interaction the sufficient condition of stability, superstability and strong superstability is also given. This criterion refers to the situation, when the pair interaction is supposed to be stable, superstable or strong superstable and the parts of the total energy contributed by $p$-body potentials (p>2) are positive and decreasing. It takes into account the traditional concept, that in some sense p-body potential plays less importantpart in the energy of interaction than p-1-one. While proving this result the new lower bound for the part of the total energy, which is defined only by $p$-body potential, is proposed. Several new notions are pre-sented to formulate this criterion. They can be used either to prove other similar criteria or to estimate the coefficients in the conditions of superstability or strong superstability. There has been investigated an example of the family of many-body potentials, that ensures superstable interaction and has immediate application in molecular physics. It generalizes the well-known Lenard--Jones pair potential on many-body case. The one-dimensional case is considered separately. Some obtained estimates in this case are exacter than in many-body situation. The exact values of the coefficients in this example are proposed. These sufficient criteria of stability, superstability and strong superstability can be used in the case of classical equilibrium continuous system to construct Gibbs measure either in finite volume or on the space of infinite configurations, when the interaction is defined both the pair potential or the family of many-body potentials. The totally new concept of quasilattice approximation of continuous systems for a pressure, correlation functions in the fixed volume and correlation functions of infinite systems is introduced. The main point of this approximation consists in the idea, that in the expressions for the basic characteristics of the system integration is carried out not over all space of configurations, but only over those configurations, which contain for the given partition of the space into hypercubes not more than one particle in each cube. The equations of Kirkwood--Salzburg type for the family of approximated correlation functions are obtained and the question about the existence of its solutions in the form of convergent in one of the specially defined spaces series is analyzed. It is proved, that if the condition of strong superstability holds, then these recently defined functions approximate with any accuracy the classical pressure, correlation functions in the fixed volume and correlation functions of infinite systems (the last result is true only in the region of small values of a chemical activity z). The result for the correlation functions of infinite systems is proved using the method of mathematical induction, technique of Kirkwood-Salzburg equations and one technical lemma. In the theorems, which are connected with quasilattice approximation the interaction is defined only by pair potential. Such an approximation helps to generalize some results in the case of lattice gas on the case of classical continuous systems. It contains also the transition parameter, that is the length a of an edge of an arbitrary cube from the partition of the space and ensures the connection between the lattice and continuous cases. It is especially important, because there are many results in the theory of lattice systems and very few for continuous ones.
