Currently countries of the world are forming legal norms that regulate the turnover of new digital objects of rights that are called differently as digital rights, tokens, digital assets, digital currency, and cryptocurrency. The difference in wording does not allow countries to develop common international approaches to the cross-border turnover of such new objects of rights. The purpose of research is to propose the model of legal regulation that would allow such objects of rights to be fully included in the civil circulation. The following research tasks are being solved: analysis of the legal norms that regulate the turnover of digital rights and digital currency; formulation of the models of legal regulation of the turnover of digital rights and digital currency; study of measures and means of legal regulation; analysis of different points of view of researchers on the issue of legal regulation of relations in the digital economy; proposal of measures and means of legal regulation, based on the chosen model of legal regulation. In carrying out the study, comparative legal, formal legal, legal modeling methods were implemented. General research methods of synthesis, analysis, induction, comparison, etc. were used. Approaches used in jurisdictions differ both in terms of adopted legal norms and in creation of special conditions for the functioning digital market. Countries use a prohibitive model of legal regulation of the turnover of digital rights and digital currency (prohibition of their issuance and turnover), partially prohibitive (restrictions on the turnover), partially permissive (admission of turnover, subject to certain conditions — licensing, regulatory sandboxes, etc.), and permissive model (allowing the turnover, subject to minimum requirements). From the experience of legal regulation of foreign countries, the attention of the legislator should be drawn to the need and possibility of licensing in relation to participants in the digital market, as well as to the successful experience of the functioning of regulatory sandboxes. When establishing law enforcement practice in the Russian Federation regarding the turnover of digital rights and digital currency, the existing experience in such jurisdictions as the United States, Great Britain, Australia as well as the effective legal regulation of the crypto industry in Japan shall be considered.