On the basis of the systemic and synergistic approaches, the features and regularity of the selforganizing systems, it is concluded that the social systems are nonlinear, open, complex, self-organizing systems. The scheme-model of the general structure of the social system is considered. Much attentions is given to the interactions of the social systems elements with the biosystems. The graphic model-tree, reflecting the association between the social systems and the biosystems. In connection with that the most significant technologies and the scientific trends, connecting with the biosystems: the gens, the genetic engineering, the artificial intellect, the bio- and nanotechnologies, the biomedicine, microbes, viruses are considered. The futures of the dynamics of the social systems are noted. It is shown, that, with a high probability, the spiral evolutionary processes can occur in the social systems. The mathematical model is presented, that describes the spiral structures, based on the Swift-Hohenberg equation.