Chapter 5 The Economic Theory of Gift Giving: Perfect Substitutability of Transfers and Redistribution of Wealth This chapter reviews the theory of the voluntary public and private redistribution of wealth elaborated by economic analysis in the last forty years or so. The central object of the theory is altruistic gift giving construed as benevolent voluntary redistribution of income or wealth. The theory concentrates on lump sum voluntary transfers individual or collective which aim at equalizing the distribution of wealth from altruistic reasons or sentiments (perfectly substitutable altruistic transfers). It implies: (i) the Pareto inefficiency of the non cooperative interaction of individual altruistic transfers; (ii) the neutralization of public transfers by individual altruistic transfers; (iii) and the crowding out of private altruistic transfers by Pareto efficient public redistribution. The chapter is organized as follows. Section 2 presents an informal overview of the general intent and content of the theory. Section 3 gives a first formal version of the theory in a one commodity setup (pure distributive social system). Non cooperative distributive equilibrium is characterized and its fundamental properties of existence and determinacy are analyzed. Section 4 extends the definitions and fundamental properties of pure distributive social systems to general social systems that combine competitive market exchange with the non cooperative altruistic transfers of individuals endowed with non paternalistic interdependent preferences. Section 5 states the neutrality property in two versions of the theory successively: the general social systems of Section 4; and the important special case of the pure distributive social systems of Section 3 where the set of agents is partitioned in two subsets namely a subset of poor individuals with zero endowments and egoistic preferences and a subset of rich individuals altruistic to the poor and indifferent to each other. Section 6 reviews the theory of Pareto efficient redistribution in pure distributive social systems. Section 7 returns to the fundamental assumption of perfect substitutability of transfers through a selective review of theoretical models of imperfectly substitutable transfers and empirical tests of perfect substitutability.