This chapter analyzes lexicographic maximization (leximax) as a fairness criterion that maxinmizes the smallest utility, then the second smallest, and so forth. Leximax fairness generalizes the Rawlsian maximin criterion and avoids some of its shortcomings, such as the fact that a maximin solution can leave available resources used. The chapter describes the socially optimal solution subject to a budget constraint both with and without utility bounds; in the latter case, it is the same as the maximin solution. Since leximax fairness is not represented by a social welfare function in the usual sense, the chapter generalizes the concept of monotone separability to show that the leximax problem is regionally decomposable.