Inverse reinforcement learning (IRL), also known as inverse optimal control (IOC), is a subfield of imitation learning and reinforcement learning. In order to learn expert behavior, IRL methods infer a reward function from expert demonstrations, then, IRL methods adopt a reinforcement learning algorithm to find out the desired behavior. In recent years, IRL methods have received a lot of attention and have been successfully used in solving a variety of tasks, such as navigation for vehicle investigation, planning trajectory, and robotic optimal control. First, the fundamental theories that include the formal definition of IRL are presented. Then, we introduce the research progress of IRL methods which include algorithms based on linear reward function and non-linear reward function, such as maximum margin approaches and maximum entropy approaches. In addition, from frontier research directions of inverse reinforcement learning, we introduce and analyze representative algorithms in this IRL which include incomplete expert demonstrations IRL approach, multi-agent IRL approach, sub-optimal expert demonstrations IRL approach, and guiding IRL approach. Finally, we summary some primary challenges and future developments in inverse reinforcement learning methods.