The growing use of aerial user equipments (UEs) in various applications requires ubiquitous and reliable connectivity for safe control and data exchange between these devices and ground stations. Key questions that need to be addressed when planning the deployment of aerial UEs are whether the cellular network is a suitable candidate for enabling such connectivity and how the inclusion of aerial UEs might impact the overall network efficiency. This paper provides an in-depth analysis of user and network-level performance of a cellular network that serves both unmanned aerial vehicles (UAVs) and ground users in the downlink. Our results show that the favorable propagation conditions that UAVs enjoy due to their height often backfire on them, as the increased load-dependent co-channel interference received from neighboring ground base stations (BSs) is not compensated by the improved signal strength. When compared with a ground user in an urban area, our analysis shows that a UAV flying at 100 m can experience a throughput decrease of a factor 10 and a coverage drop from 76% to 30%. Motivated by these findings, we develop UAV and network-based solutions to enable an adequate integration of UAVs into cellular networks. In particular, we show that an optimal tilting of the UAV antenna can increase the coverage from 23% to 89% and throughput from 3.5 to 5.8 b/s/Hz, outperforming ground UEs. Furthermore, our findings reveal that depending on the UAV altitude and its antenna configuration, the aerial user performance can scale with respect to the network density better than that of a ground user. Finally, our results show that network densification and the use of microcells limit the UAV performance. Although UAV usage has the potential to increase the area spectral efficiency (ASE) of cellular networks with a moderate number of cells, they might hamper the development of future ultradense networks.