Unmanned aerial vehicles (UAVs) have found numerous applications and are expected to bring fertile business opportunities in the next decade. Among various enabling technologies for UAVs, wireless communication is essential and has drawn significantly growing attention in recent years. Compared to the conventional terrestrial communications, UAVs’ communications face new challenges due to their high altitude above the ground and great flexibility of movement in the 3-D space. Several critical issues arise, including the line-of-sight (LoS) dominant UAV-ground channels and induced strong aerial-terrestrial network interference, the distinct communication quality-of-service (QoS) requirements for UAV control messages versus payload data, the stringent constraints imposed by the size, weight, and power (SWAP) limitations of UAVs, as well as the exploitation of the new design degree of freedom (DoF) brought by the highly controllable 3-D UAV mobility. In this article, we give a tutorial overview of the recent advances in UAV communications to address the above issues, with an emphasis on how to integrate UAVs into the forthcoming fifth-generation (5G) and future cellular networks. In particular, we partition our discussion into two promising research and application frameworks of UAV communications, namely UAV-assisted wireless communications and cellular-connected UAVs, where UAVs are integrated into the network as new aerial communication platforms and users, respectively. Furthermore, we point out promising directions for future research.