This paper studies the market for monopolistically supplied sweepstakes. We derive equilibrium demands for fixed-prize and variable-prize sweepstakes and determine the profit-maximizing prize level and pay-out ratio respectively. It can be profitable to offer each type of sweepstake when there is a large enough number of weighted utility consumers who have constant absolute risk attitudes, are strictly averse to small as well as symmetric risks, and display longshot preference behaviour. Moreover, for the variable-prize sweepstake, the supplier will generally find it profitable to combine sweepstakes targeting two smaller populations, and offer a single sweepstake to the combined population. This implication is corroborated by the recent spate of mergers of smaller state lotteries into larger ones. Copyright The Review of Economic Studies Limited, 2005.