We survey the theories on why banks promise to pay par on demand and examine evidence on the conditions under which banks have promised to pay the par value of deposits and banknotes on demand when holding only fractional reserves. The theoretical literature is divided into four strands: liquidity provision; asymmetric information; legal restrictions; and a medium of exchange. We assume that it is not zero cost to make a promise to redeem a liability at par value on demand. If so, then the conditions in the theories that result in par redemption are possible explanations why banks promise to pay par on demand. If the explanation based on customers’ demand for liquidity is correct, payment of deposits at par will be promised when banks hold assets that are illiquid in the short run. If the asymmetric-information explanation based on the difficulty of valuing assets is correct, the marketability of banks’ assets determines whether banks promise to pay par. If the legal restrictions explanation of par redemption is correct, banks will not promise to pay par if they are not required to do so. If the transaction explanation is correct, banks will promise to pay par if the deposits are used in transactions. We examine the history of banking in several countries in different eras: fourth century Athens, medieval Italy, Tokugawa Japan, and free banking and money market mutual funds in the United States. Each of the theories explains some of the observed banking arrangements and none explains all of them.