The rapid growth of consumer Unmanned Aerial Vehicles (UAVs) is creating promising new business opportunities for cellular operators. On the one hand, UAVs can be connected to cellular networks as new types of user equipment, therefore generating significant revenues for the operators that can guarantee their stringent service requirements. On the other hand, UAVs offer the unprecedented opportunity to realize UAVmounted flying base stations that can dynamically reposition themselves to boost coverage, spectral efficiency, and user quality of experience. Indeed, the standardization bodies are currently exploring possibilities for serving commercial UAVs with cellular networks. Industries are beginning to trial early prototypes of flying base stations or user equipments, while academia is in full swing researching mathematical and algorithmic solutions to address interesting new problems arising from flying nodes in cellular networks. In this article, we provide a comprehensive survey of all of these developments promoting smooth integration of UAVs into cellular networks. Specifically, we survey (i) the types of consumer UAVs currently available off-the-shelf, (ii) the interference issues and potential solutions addressed by standardization bodies for serving aerial users with the existing terrestrial base stations, (iii) the challenges and opportunities for assisting cellular communications with UAV-based flying relays and base stations, (iv) the ongoing prototyping and test bed activities, (v) the new regulations being developed to manage the commercial use of UAVs, and (vi) the cyber-physical security of UAV-assisted cellular communications.