The European Central Bank will soon have to decide on her monetary strategy. This paper discusses properties of money demand functions and implications for monetary policy in a monetary targeting regime. Special attention is paid to different concepts of stability of money demand, to the length and variability of adjustment lags, and to the controllability of the money stock. At the European level, stable money demand functions are identified for M1 and M3. However, M3 appears to be less controllable than M1. Long and variable adjustment lags in the demand for M3 weaken the case for broad money as a monetary target.