This thesis focuses on the area of solutions for TU-games, which contains new axiomatic characterizations of either solutions already studied in the literature or new solutions. The thesis consists of six chapters. Chapter 1 introduces fundamental terminologies and notations. Chapter 2 defines and characterizes a new solution for TU-games, namely the average-surplus value. Firstly, we define the average-surplus value by an underlying procedure of sharing this marginal surplus. Then, we characterize the average-surplus value by introducing the A-null surplus player property and revised balanced contributions. Next, we define the AS-potential function, and show that the adjusted marginal contributions vector of the AS-potential function coincides with the average-surplus value. Finally, we provide a non-cooperative game, and show that the outcome in every subgame perfect equilibrium of this game coincides with the payoff assigned by the average-surplus value. Chapter 3 provides new axiomatic characterizations of the EANSC value and the CIS value. Firstly, we introduce an alternative way to reevaluate the worth by considering players in the coalition as a whole, and define the E-union associated game and the C-union associated game. Then, adopting E-union associated consistency and C-union associated consistency, we provide new axiomatizations of the EANSC value and the CIS value. Finally, we propose two dynamic processes on the basis of these associated games that lead to the CIS value and EANSC value. This follows from a more general result showing that these dynamic processes can lead to any solution satisfying the inessential game property and continuity. Chapter 4 presents characterizations of the PD value and the PANSC value. Firstly, we define the optimistic satisfaction and pessimistic satisfaction, and show that the PD value and the PANSC value can be obtained by maximizing the minimal optimistic satisfaction and pessimistic satisfaction, respectively, in the lexicographic order over the non-empty pre-imputation set. Secondly, we characterize the PD value and the PANSC value by introducing equal minimal optimistic satisfaction and equal minimal pessimistic satisfaction. Thirdly, we also characterize the PD value and the PANSC value by introducing optimistic associated consistency and pessimistic associated consistency. Finally, we define the dual axioms of the optimistic associated consistency and pessimistic associated consistency axioms, and characterize these two proportional values on the basis of these dual axioms. Chapter 5 studies axiomatic foundations of the class of weighted division values. Firstly, we consider relaxations of symmetry, specifically sign symmetry and weak sign symmetry, to characterize the class of (positively) weighted division values. Secondly, we show that the class of weighted division values can also be characterized by replacing linearity in the three axiomatizations of B\'{e}al et al. (2016) with additivity. Finally, we show how strengthening an axiom regarding null, non-negative, respectively nullified players in these above axiomatizations, provides three axiomatizations of the positively weighted division values. Chapter 6 turns to an application, specifically pollution cost-sharing problems. We explore how to share the cost of cleaning up a polluted river using cooperative game theory, and study axiomatic foundations of cost-sharing methods for pollution cost-sharing problems. Firstly, we suggest a relaxation of independence of upstream costs, to characterize the UES method. Secondly, we propose the classes of EUR methods and WUS methods, which generalize the LRS method and the UES method. We provide two axiomatizations of the class of EUR methods, one using this weak independence axiom and one using a weak version of the no blind cost axiom. Meanwhile, we provide two axiomatizations of the class of WUS methods by introducing two weak versions of upstream symmetry. Finally, we define a pollution cost-sharing game, and show that the compromise method coincides with applying the Shapley value to this game.