This chapter is dedicated to the study of distributed resource allocation problems in wireless communication networks. Non-cooperative game theory proves to be a useful tool to investigate this type of problems. The players, the transmitter nodes, choose their power allocation policies in order to maximize their own information theoretic payoffs, namely their individual achievable Shannon transmission rates. Our attention is mostly focused on the basic multi-user channel models: the multiple access channel (MAC) and the interference channel (IC). However, more complex channels such as the interference relay channel (IRC) and the cognitive radio channel (CRC) are also discussed. We provide an updated overview of the existing results with respect to the non-cooperative solution of the game, the Nash equilibrium (NE), its existence, uniqueness, and convergence of distributed algorithms. Furthermore, we evaluate the performance gap between the distributed and the centralized approach where a network authority allocates the overall resources of the network. We discuss several methods that improve the performance of the NE at the cost of introducing a supplementary signaling cost or intervention of a central authority or a certain cooperation degree at the user level. We conclude by a critical discussion about the drawbacks and possible improvements of the game theoretical approach to solve resource allocation problems in general.