Exchange is the essence of economics, and there is a large literature extolling the virtues of using money as a means of exchange over other forms of exchange. Despite this, many trades do not take the form of goods for cash, even in instances where money could conceivably be used. For instance, individuals provide favors for one another, with the promise of a future reciprocated favor acting as implicit payment. Similarly, there is considerable evidence from sociology and anthropology that there is a marked reticence to use money in social relations. This proposal considers why many trades remain non-monetized even in modern monetized economies. We consider three separate aspects of non-monetary trade in this proposal, namely, (i) the role of money in reciprocal exchange, (ii) the use of money in social situations, and (iii) the use of commercial barter exchanges. In each of these settings, we illustrate novel reasons why agents may prefer to avoid using fiat money despite its usual advantages in achieving a double coincidence of wants, and may instead employ non-monetary exchange. By addressing the importance of non-monetary exchange for reciprocity, payment-in-kind, and barter markets, we develop a richer understanding of how agents trade, the most basic ingredient of economics.