Chapter 13 Axiomatizations of the core The core is the most intuitive solution concept in cooperative game theory. An intuitively acceptable axiom system for the core might reinforce its position as the most natural solution. An axiomatization of the core may serve two other more important goals: (1) by obtaining axioms for the core those important properties of solutions are singled out that determine the most stable solution in the theory of cooperative games. Thus the core of transferable utility (TU) games is determined by individual rationality (IR) superadditivity (SUPA) and the reduced garne property (RGP). Also the core of non transferable utility (NTU) garnes is characterized by IR and RGP. Furthermore the converse reduced game property (CRGP) is essential for the axiomatization of the core of TU market games. Four properties IR SUPA RGP and CRGP play an important role in the characterization of the core on some important families of games. A solution is acceptable if its axiomatization is similar to that of the core. There are some important examples of this kind: (a) the prenucleolus is characterized by RGP together with the two standard assumptions of symmetry and covariance (b) the Shapley value is characterized by SUPA and three more weaker axioms and (c) the prekernel is determined by RGP CRGP and three more standard assumptions. The chapter discusses the TU games several properties of solutions to coalitional games an axiomatization of the core of balanced games the core of market games the results that are generalized to games with coalition structures the results for NTU games reduced games of NTU games axiom system for the core of NTU games and Keidings axiomatization of the core of NTU games.