The rise of new forms of private money is reviving a long-standing debate on the appropriate balance between private and public interests in money and payments. The main aim of this paper is to explore an integrated policy analysis of various digital assets that may function as money: bank deposits, non-backed crypto’s, stablecoins and Central Bank Digital Currency (CBDC). In our view, public and private money should coexist to get the best of both worlds: trust and innovation. Getting the balance right is however not an easy task. It requires a digital update of public money and effective regulation of crypto’s and stablecoins. We argue that convertibility between public and private money should be a leading principle both for the design of CBDC and for the regulation of stablecoins that could potentially be widely adopted as a means of payment.