Urban Air Mobility (UAM) is an emerging transport mode, offering on-demand and automated air transit of passengers or cargo within urban ecosystems. Unrestricted by terrestrial road networks, UAM provides a timely and efficient way of transportation. Due to its profound significance, industries, regulatory authorities, and academic researchers have made extensive efforts toward its development and improvement. This paper contributes to these ongoing endeavors by providing a comprehensive and systematic review of existing literature on UAM from the perspective of airspace and traffic management. Specifically, this paper chronologically encapsulates the development history of UAM, aiming to foster a holistic understanding of contributions made by industries, academics, and regulatory authorities. It also highlights specific advances such as the integration of AI in traffic flow algorithms and the development of collision avoidance systems, addressing current technological and regulatory hurdles. Furthermore, this paper provides an in-depth review of the studies on the topic of urban air transportation network design, urban air traffic flow management, and urban flight safety management. The diverse scenarios and methodologies employed across those studies are discussed and compared, identifying gaps in current research and potential for future exploration. Lastly, this paper highlights opportunities and prospective directions for future research in relation to traffic management strategies, thus propelling UAM toward greater efficiency. rban Air Mobility (UAM) is an emerging transport mode, offering on-demand and automated air transit of passengers or cargo within urban ecosystems. Unrestricted by terrestrial road networks, UAM provides a timely and efficient way of transportation. Due to its profound significance, industries, regulatory authorities, and academic researchers have made extensive efforts toward its development and improvement. This paper contributes to these ongoing endeavors by providing a comprehensive and systematic review of existing literature on UAM from the perspective of airspace and traffic management. Specifically, this paper chronologically encapsulates the development history of UAM, aiming to foster a holistic understanding of contributions made by industries, academics, and regulatory authorities. It also highlights specific advances such as the integration of AI in traffic flow algorithms and the development of collision avoidance systems, addressing current technological and regulatory hurdles. Furthermore, this paper provides an in-depth review of the studies on the topic of urban air transportation network design, urban air traffic flow management, and urban flight safety management. The diverse scenarios and methodologies employed across those studies are discussed and compared, identifying gaps in current research and potential for future exploration. Lastly, this paper highlights opportunities and prospective directions for future research in relation to traffic management strategies, thus propelling UAM toward greater efficiency.