Abstract: The hierarchical equations of motion (HEOM) formalism developed recently is becoming an important method for investigating the properties and dynamical processes of quantum open systems. A comprehensive account on its establishment and recent development was given, covering theoretical formulations, numerical algorithm and practical applications. By construction HEOM resolves in a nonperturbative manner the combined effects of many-body interaction, system-environment coupling and non-Markovian memory. While it is mathematically equivalent to an exact path integral formalism, HEOM is also very convenient in computation and versatile in various systems of study. Moreover, it is shown recently that HEOM is actually a correlated system-environment dynamics theory. The practicality of HEOM to address physical and chemical problems was exemplified with examples relevant to quantum dissipation and quantum transport.

Key words: quantum dissipation, quantum transport, open system, system-bath coherence