With the emergence of the Internet-of-Things (IoT), artificial intelligence, and communication technologies, cyber-physical systems (CPS) have revolutionized the engineering paradigm with profound applications in many aspects of society including homes, energy, agriculture, health-care, transportation, business, and manufacturing. A CPS uses suitable computational techniques such as game theory to enable different entities to interact with one another for taking necessary actions to obtain selected objectives. Recent literature on CPS has extensively used game theory to approach a variety of technical challenges. In order to make these contributions more accessible to a broader audience, there is a need for studies that can provide readers with a comprehensive understanding of different types of CPS and their attributes, then clearly outline why game theory is relevant for modeling different aspects of CPS, and also discuss how game theory has been used in relevant literature to date. This paper bridges this gap by 1) providing a general discussion of different types of CPS and their characteristics; 2) giving an overview of different types of game-theoretic approaches; 3) explaining why game theory is appropriate for modeling different types of CPS; and 4) finally, studying how game theory has been used in different CPS types to address their challenges. Further, we also identify some key research challenges for future investigation where game theory could be applied as a potential solution.