Unmanned AerialVehicles (UAVs) equipped with air quality sensors offer a powerful solution for increasing the spatial and temporal resolution of air quality data, searching and detecting emission sources, and monitoring emissions from fixed and mobile sources. Despite the numerous advantages of using UAVs, their use, however, presents several challenges that limit their broader adoption. For example, UAVs require efficient algorithms and components to minimize power consumption, the overall payload used on UAVs needs to be small to ensure optimal portability which poses limitations on the sensors that can be integrated with UAVs, and there is a need for specialized algorithms, e.g., for identifying and locating air pollution sources. Currently, most solutions for UAV-based air quality monitoring focus on specific challenges or demonstrating the potential of using UAVs, and there is a lack of comprehensive overview of the research field and its open challenges. In this paper, we contribute a systematic review of UAV-based air quality monitoring, highlighting and analyzing technical solutions and challenges, and identifying open challenges with the aim of providing a research roadmap for the path forward.