The article discusses social potential of civil society and the possibilities of its implementation, associated with the process of institutionalization of civil society. Such forms of public activity as volunteering and charity, supported by broad groups of citizens, are considered. The dependence of volunteerism on the nature of civil responsibility is demonstrated. Calculation formulas and values of indices of the mutual influence of civil responsibility and volunteering, as well as of indices of the influence of civil responsibility on involvement in charity are presented. The analysis carried out and the opinion of a certain part of experts allows to conclude that it is possible to form a «solidarity society» characterized by a high level of the population activity; however, the majority of experts believe that the involvement of citizens in the established institutions of civil society is at medium and low levels. The authors, using the materials of the discussion between S.G. Kirdina and A.G. Myasnikov on the essence of the concept of «civil society» in Russian sociology, propose a model of modern civil society in Russia, which involves three elements: civic participation, civil society, and civic influence. In this case, civic participation is understood as the readiness and ability of individuals, social groups and communities to participate in public activities, to establish norms and rules for social forms of behavior and relations, which means the process of institutionalization of civil society. Whereas the intensity and productivity of this process is determined by the ability and possibility of civic influence, i.e. the ability and possibility to influence the opinions and behavior of individuals, groups, communities and society.