The paper studies a new class of games, "All-Pay Contests", which capture general asymmetries and sunk investments inherent in scenarios such as lobbying, competition for market power, labor-market tournaments, and R&D races. Players have continuous, non-decreasing cost functions and compete for one of several identical prizes. The generality of players' cost functions allows for differing production technologies, costs of capital, and prior investments, among others. I provide a closed-form formula for players' equilibrium payoffs, and use it to compute aggregate expenditures, derive the effects of changes in contest structure, and analyze player participation. An algorithm for computing the unique equilibrium is given for a subclass of contests. This subclass nests multi-prize, complete-information all-pay auctions.