Lecture on the first SFB/TR 15 meeting, Gummersbach, July, 18 - 20, 2004: We analyze the allocation of priority in queues via simple bidding mechanisms. In our model, the stochastically arriving customers are privately informed about their own processing time. They make bids upon arrival at a queue whose length is unobservable. We consider two bidding schemes that differ in the definition of bids (these may reflect either total payments or payments per unit of time) and in the timing of payments (before, or after service). In both schemes, a customer obtains priority over all customers (waiting in the queue or arriving while he is waiting) who make lower bids. Our main results show how the convexity/concavity of the function expressing the costs of delay determines the queue-discipline (i.e., SPT, LPT) arising in a bidding equilibrium.