Wireless communication is an essential technology to unlock the full potential of unmanned aerial vehicles (UAVs) in numerous applications and has thus received unprecedented attention recently. Although technologies such as direct link, WiFi, and satellite communications are still useful in some remote scenarios where cellular services are unavailable, it is believed that exploiting the thriving 5G and beyond cellular networks to support UAV communications is the most promising and cost-effective approach, especially when the number of UAVs grows dramatically. On the one hand, to guarantee safe and efficient flight operations of multiple UAVs, it is of paramount importance to provide secure and ultra-reliable communication links between the UAVs and their ground pilots or control stations for conveying command and control signals, especially in beyond-visual-line-of-sight (BVLOS) scenarios. On the other hand, because of advances in communication equipment miniaturization as well as UAV manufacturing, mounting compact and lightweight base stations (BSs) or relays on UAVs becomes increasingly feasible. This has led to two promising research paradigms for UAV communications, namely, UAV-assisted cellular communications and cellular-connected UAVs, where UAVs are integrated into cellular networks as aerial communication platforms and aerial users, respectively. As such, integrating UAVs into cellular networks is believed to be a win-win technology for both UAV-related industries and cellular network operators, which not only creates plenty of new business opportunities but also benefits the communication performance of 3-D wireless networks. In addition, UAV related sensing and computing are also helpful for achieving efficient and reliable communication (e.g., in avoiding coverage holes) as well as smart UAV coordination, positioning, and trajectory design. However, 5G and beyond wireless networks with UAVs significantly differs from traditional communication systems, because of the high altitude and high maneuverability of UAVs, the unique UAV-ground channels, the diversified quality of service (QoS) requirements for downlink command and control (C&C) and uplink mission-related data transmission, the stringent constraints imposed by the size, weight, and power (SWAP) limitations of UAVs, as well as the new design degrees of freedom enabled by joint UAV mobility control and communication resource allocation.