"Random-matching models" in monetary econmomics consider a very large population (mathematically, a nonatomic measure space) of individuals who, at each date t = 0, 1, 2..., are randomly matched into pairs. Each individual carries money, the holding of which at each date is a natural number. In each pair of individuals at each date, one is randomly selected to transfer one unit of money to the other--if he possesses a unit to transfer. Is there a stationary distribution of individuals' money holdings in this population, and if so, is it approached from an arbitrary initial distribution? The answer is that a money-holdings distribution is stationary iff it is geometric, and that (under a mild assumption) an initial distribution converges to the geometric distribution with the same expectation. Besides a presentation of this result (due to E. Green and R. Zhou, Econometrica, 2002), the talk will include a discussion of what sort of generalization would be helpful to monetary economists.