A survey of max-type recursive distributional equations

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form Xd=g((.i,Xi),i.1). Here (.i) and g(.) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(.) is essentially a .maximum. or .minimum. function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process Xi, are the Xi measurable functions of the innovations process (.i)?