At the root of the notion of stablecoin (SC) lies a desire to reconcile two different worlds: that of legal currency, whose essential attributes are hierarchical order, the vocation to uniqueness and stability of the purchasing power, and that of crypto-assets, featuring decentralization, multiplicity and thus the possibility of choice, and the instability of value. Do SCs fulfill their promises? With regard to their volatile prices, limited number, small total amount, and concentrated market, SCs have so far met with a mixed success. They rather represent a complement to the crypto-assets market. However, the arrival of very large issuers, securing a higher degree of confidence to users, and apt to reach a wide public, could give their projects a potentially systemic impact. These global SCs would create risks, in particular for financial stability and monetary policy, and in lesser-developed economies. This paper reviews these risks and the way the private sector, regulators and central banks can address them.