The human action is an important information source for human social interaction, and it simultaneously plays a crucial role in human-robot interaction (HRI). For a natural and fluent interaction, robots are required to understand human actions and have the capacity to predict action intentions and to imitate human actions for an appropriate response. Currently, existing survey papers for the action recognition mainly summarize algorithms that perform action recognition in experimental scenarios, and survey papers of the HRI mainly introduced various interaction interfaces in the HRI. Different from these surveys, we focus on the human action analysis on robot platforms for the HRI application, including the body motion and gestures. We review the existing HRI related references involving the action recognition, prediction, and the robot imitation of the human action. Moreover, we give a summary of robot platforms and action datasets that are frequently used in the study of HRI. Finally, we give an analysis on the development trend and future research directions of action analysis for the HRI applications.