Abstract: Science exploration based on increasingly sophisticated artificial intelligence has witnessed great technological advances． Meanwhile, mathematical foundation of intelligent phenomena has emerged as a crucial scientific issue requiring urgent resolution． Of the numerous artificial intelligence algorithms, reservoir computing, due to its simple structure, flexible and diverse implementation, and characteristics of nonlinearity and memory storage, is widely applied in the research of time series and dynamical system-related problems． Trained reservoir forms a complex dynamical system． Serving as a bridge, analysis by dynamical system theory through the reservoir plays an important role in exploring intelligent phenomena in the learning process． The latest progress in reservoir computing algorithm is reviewed in this work, outlining two complementary research directions: reservoir computing for dynamical systems, dynamical system foundations of reservoir computing． The present work will deepen understanding of artificial intelligence algorithms, and promote interdisciplinary research and communication among artificial intelligence, complex systems, and statistical physics-related disciplines．