Inverse reinforcement learning (IRL), also known as inverse optimal control (IOC), is an important research method of reinforcement learning and imitation learning. IRL solves a reward function from expert samples, and the optimal strategy is then solved to imitate expert strategies. In recent years, fruitful achievements have been yielded by IRL in imitation learning, with widespread application in vehicle navigation, path recommendation, and robotic optimal control. First, this study presents the theoretical basis of IRL. Then, from the perspective of reward function construction methods, IRL algorithms based on linear and non-linear reward functions are analyzed. The algorithms include maximum marginal IRL, maximum entropy IRL, maximum entropy deep IRL, and generative adversarial imitation learning. In addition, frontier research directions of IRL are reviewed to compare and analyze relevant representative algorithms containing IRL with incomplete expert demonstrations, multi-agent IRL, IRL with sub-optimal expert demonstrations, and guiding IRL. Finally, the primary challenges of IRL and future developments in its theoretical and application significance are summarized.