The widespread application of networks is providing a better platform for the development of society and technology. With the expansion of the scope of network applications, many issues need to be solved. Among them, the maximization of utility and the improvement of security have attracted much attention. Many existing attacks mean the network faces security challenges. The concept of trust should be considered to address these security issues. Meanwhile, the utility of the network, including efficiency, profit, welfare, etc., are concerns that should be maximized. Over the past decade, the concepts of game and trust have been introduced to various types of networks. However, there is a lack of research effort on several key points in distributed networks, which are critical to the information transmission of distributed networks, such as expelling malicious nodes quickly and accurately and finding equilibrium between energy assumption and high transmission rate. The purpose of this paper is to give a holistic overview of existing research on trust and game theory in networks. We analyzed that network utility can be maximized in terms of effectiveness, profits, and security. Moreover, a possible research agenda is proposed to promote the application and development of game theory and trust for improving security and maximizing utility.