This review presents and reviews various solved and open problems in developing, analyzing, and mitigating epidemic spreading processes under human decision-making. We provide a review of a range of epidemic models and explain the pros and cons of different epidemic models. We exhibit the art of coupling between epidemic models and decision models in the existing literature. More specifically, we provide answers to fundamental questions in human decision-making amid epidemics, including what interventions to take to combat the disease, who are decision-makers, and when and how to take interventions, and how to make interventions. Among many decision models, game-theoretic models have become increasingly crucial in modeling human responses or behavior amid epidemics in the last decade. In this review, we motivate the game-theoretic approach to human decision-making amid epidemics. This review provides an overview of the existing literature by developing a multi-dimensional taxonomy, which categorizes existing literature based on multiple dimensions, including (1) types of games, such as differential games, stochastic games, evolutionary games, and static games; (2) types of interventions, such as social distancing, vaccination, quarantine, and taking antidotes; (3) the types of decision-makers, such as individuals, adversaries, and central authorities at different hierarchical levels. A fine-grained dynamic game framework is proposed to capture the essence of game-theoretic decision-making amid epidemics. We showcase three representative frameworks with unique ways of integrating game-theoretic decision-making into the epidemic models from a vast body of literature. Each of the three frameworks has their unique way of modeling and analyzing and develops results from different angles. In the end, we identify several main open problems and research gaps left to be addressed and filled.