The emergence of private digital currencies poses a threat to payment systems and monetary policy because they challenge all functions of money as we know them. In this paper, we focus mainly on the banking and monetary policy issues raised by stablecoins and Central Bank Digital Currencies (CBDCs). We begin by describing the current working of bank-centered payment systems. We next touch upon cryptoassets and focus on the domestic and international impact of stablecoins. We identify two problems with stablecoins: their role in the breakup of uniform currency, and as a source of financial instability due to the lack of a monetary backstop in the event of a run to withdraw funds in adverse situations. We then deal with the pros and cons of CBDCs, whether they are an adequate response to the challenges raised by stablecoins, their possible impact on monetary and banking policy, and some open economy issues. While we do not see major advantages in private digital currencies as a payment system or as an investment, we also do not find any robust motivation for the introduction of CBDCs beyond geopolitical reasons and the oversight of a technological area.