Suppose that n cars C_1, ..., C_n want to park on a one-way street with n parking spaces 1,2,...,n. Each car has a preferred space a_i. The cars go one at a time in the order C_1, ..., C_n to their preferred space and then park in the first available space. If all the cars can park the the sequence (a_1,...,a_n) is called a parking function. For instance, the three parking functions of length two are (1,1), (1,2), and (2,1).

his talk will be a survey of the theory of parking functions. Topics include (1) the characterization and enumeration of parking functions, (2) connections with tree inversions, noncrossing partitions, and the Shi hyperplane arrangement, and (3) a convex polytope closely associated with parking functions.