As the most abundant element in the universe, hydrogen is a promising energy carrier for decarbonizing various economic sectors. Green hydrogen production from water electrolysis is critical to the success of this path with polymer electrolyte membrane (PEM) water electrolyzer (WE) as a key technology due to its quick dynamic response and high energy efficiency. Nevertheless, vigorous control algorithms are necessary to maximize the performance, efficiency, and useable lifetime of PEM WEs. This review attempts to collate the modeling frameworks relevant to controller design and provides a survey of various control techniques used in literature to overcome the challenges associated with the transient operation of PEM WEs. To better understand the underlying physics and the coupling between different subsystems, we first review control-oriented electrochemical, thermal, mass transport, and equivalent circuit models. We identify manipulable system variables and control knobs that can be employed for a better system operation in the next step, and finally, we discuss different controllers used in literature, including traditional control approaches, optimal control methods, and other advanced techniques such as nonlinear and neural network controllers.