This article explains the theoretical importance of the quality of money as a factor of the demand for money and develops the composite indicator that measures the quality of money for the eurozone. The demand for money, i.e., the amount of money people keep in their balances, besides other well-known factors (e.g., interest rate, price level, and income) depends on how people subjectively perceive a particular money’s ability to serve its main functions: a medium of exchange, a store of value, and the unit of account. These properties depend not only on the instruments of monetary policy and the extent to which they are used, but also on the institutional framework of the monetary system. The article suggests that the quality of money is influenced by the institutional framework and monetary policy and that thus the quality of money is a separate channel for the transmission of money policy that works not through the usual mechanism of changing the supply of money, but through central banks affecting the demand for money. An important contribution of this article is that it develops an empirical composite indicator, which measures the quality of money in the eurozone in 1999–2019 and shows the gradual decline in the quality of euro.