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In the recent decades, the design and development of biomimetic micro air vehicles have gained increased interest by the global scientific and engineering communities. This has given greater motivation to study and understand the aerodynamics involved with winged insects. Dragonflies demonstrate unique and superior flight performance than most of the other insect species and birds. They are capable of sustained gliding flight as well as hovering and able to change direction very rapidly. Pairs of independently controlled forewings and hindwings give them an agile flying ability. This article presents a review of all published journal articles, listed in the Thomson-Reuters Web-of-Science database (1985–2018), that are related to the flight aerodynamics of dragonflies or micro air vehicles that biomimic them. The effects of dragonfly wing motions and interactions (between forewing and hindwing) that are necessary to generate the appropriate aerodynamic forces in different flight modes are described. The associated power requirements of these modes are also addressed. This article aims to provide a valuable reference to the aerodynamic design and control of dragonfly-inspired biomimetic micro air vehicles.