Coalitional games are games in which we observe possible coalitions between players and their potential pay-off. We assume that binding agreements exist and that payoffs can be freely distributed among the members of a coalition in any way desired. In the first section we define imputations and the concept of domination. Those are the basic concepts which we will use to define the core and domination core. They represent the possible solutions to a game: which coalitions will be formed and what will be the pay-off distribution. We show the conditions for which the core is non-empty and that the domination core is equal to the core if the worth of the grand coalition (a coalition of which every player is a member) is greater than the sum of worth of any other coalition and the rest of individual worths. Then we define simple games, we show a characterization of their core via the set of veto players and we give exact conditions in which the core of a simple game is equal to the domination core. We introduce stable sets as an alternative solution to a game, we show the connection between the stable sets and the core and we discuss the differences. We also introduce balanced sets and balanced maps. Then we prove the Bondareva-Sharpley theorem which states that the core of a balanced game is non-empty. In the second section we try to get a unique point in the core, which we will accomplish by defining prenucleolus and nucleolus via excess. We define excess as the difference between the worth of a coalition and the worth which a distribution vector assigns to the said coalition. We introduce lexicographic order as a means of comparsion of different vectors and their respective excess. Then we discuss the existence and uniqueness of the nucleolus. We show which conditions have to hold for the nucleolus to be equal to the prenucleolus. Finally, we derive the Kohlberg criterion for the prenucleolus, and show the method to compute the (pre)nucleolus based on solving a sequence of linear programs.