With the increasing maturity of unmanned aerial vehicles (UAVs), also known as drones, wireless ecosystem is experiencing an unprecedented paradigm shift. These aerial platforms are specifically appealing for a variety of applications due to their rapid and flexible deployment, cost-effectiveness, and high chance of forming line-of-sight (LoS) links to the ground nodes. As with any new technology, the benefits of incorporating UAVs in existing cellular networks cannot be characterized without completely exploring the underlying trade space. This requires a detailed system-level analysis of drone cellular networks by taking the unique features of UAVs into account, which is the main objective of this dissertation. We first focus on a static setup and characterize the performance of a three-dimensional (3D) two-hop cellular network in which terrestrial base stations (BSs) coexist with UAVs to serve a set of ground user equipment (UE). In particular, a UE connects either directly to its serving terrestrial BS by an access link or connects first to its serving UAV which is then wirelessly backhauled to a terrestrial BS (joint access and backhaul). We consider realistic antenna radiation patterns for both BSs and UAVs using practical models developed by the third generation partnership project (3GPP). We assume a probabilistic channel model for the air-to-ground transmission, which incorporates both LoS and non-LoS links. Assuming the max-power association policy, we study the performance of the network in both amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols. Using tools from stochastic geometry, we analyze the joint distribution of distance and zenith angle of the closest (and serving) UAV to the origin in a 3D setting. Further, we identify and extensively study key mathematical constructs as the building blocks of characterizing the received signal-to-interference-plus-noise ratio (SINR) distribution. Using these results, we obtain exact mathematical expressions for the coverage probability in both AF and DF relaying protocols. Furthermore, considering the fact that backhaul links could be quite weak because of the downtilted antennas at the BSs, we propose and analyze the addition of a directional uptilted antenna at the BS that is solely used for backhaul purposes. The superiority of having directional antennas with wirelessly backhauled UAVs is further demonstrated via extensive simulations. Second, we turn our attention to a mobile setup and characterize the performance of several canonical mobility models in a drone cellular network in which UAV base stations serve UEs on the ground. In particular, we consider the following four mobility models: (i) straight line (SL), (ii) random stop (RS), (iii) random walk (RW), and (iv) random waypoint (RWP), among which the SL mobility model is inspired by the simulation models used by the 3GPP for the placement and trajectory of UAVs, while the other three are well-known canonical models (or their variants) that offer a useful balance between realism and tractability. Assuming the nearest-neighbor association policy, we consider two service models for the UEs: (i) UE independent model (UIM), and (ii) UE dependent model (UDM). While the serving UAV follows the same mobility model as the other UAVs in the UIM, it is assumed to fly towards the UE of interest in the UDM and hover above its location after reaching there. We then present a unified approach to characterize the point process of UAVs for all the mobility and service models. Using this, we provide exact mathematical expressions for the average received rate and the session rate as seen by the typical UE. Further, using tools from the calculus of variations, we concretely demonstrate that the simple SL mobility model provides a lower bound on the performance of other general mobility models (including the ones in which UAVs follow curved trajectories) as long as the movement of each UAV in these models is independent and identically distributed (i.i.d.). Continuing our analysis on mobile setups, we analyze the handover probability in a drone cellular network, where the initial positions of the UAVs serving the ground UEs are modeled by a homogeneous Poisson point process (PPP). Inspired by the mobility model considered in the 3GPP studies, we assume that all the UAVs follow the SL mobility model, i.e., move along straight lines in random directions. We further consider two different scenarios for the UAV speeds: (i) same speed model (SSM), and (ii) different speed model (DSM). Assuming nearest-neighbor association policy, we characterize the handover probability of this network for both mobility scenarios. For the SSM, we compute the exact handover probability by establishing equivalence with a single-tier terrestrial cellular network, in which the BSs are static while the UEs are mobile. We then derive a lower bound for the handover probability in the DSM by characterizing the evolution of the spatial distribution of the UAVs over time. After performing these system-level analyses on UAV networks, we focus our attention on the air-to-ground wireless channel and attempt to understand its unique features. For that, we first study the impact of UAV wobbling on the coherence time of the wireless channel between UAVs and a ground UE, using a Rician multi-path channel model. We consider two different scenarios for the number of UAVs: (i) single UAV scenario (SUS), and (ii) multiple UAV scenario (MUS). For each scenario, we model UAV wobbling by two random processes, i.e., the Wiener and sinusoidal processes, and characterize the channel autocorrelation function (ACF) which is then used to derive the coherence time of the channel. For the MUS, we further show that the UAV-UE channels for different UAVs are uncorrelated from each other. One key observation that is revealed from our analysis is that even for small UAV wobbling, the coherence time of the channel may degrade quickly, which may make it difficult to track the channel and establish a reliable communication link. Finally, we develop an impairments-aware air-to-ground unified channel model that incorporates the effect of both wobbling and hardware impairments, where the former is caused by random physical fluctuations of UAVs, and the latter by intrinsic radio frequency (RF) nonidealities at both the transmitter and receiver, such as phase noise, in-phase/quadrature (I/Q) imbalance, and power amplifier (PA) nonlinearity. The impact of UAV wobbling is modeled by two stochastic processes, i.e., the canonical Wiener process and the more realistic sinusoidal process. On the other hand, the aggregate impact of all hardware impairments is modeled as two multiplicative and additive distortion noise processes, which is a well-accepted model. For the sake of generality, we consider both wide-sense stationary (WSS) and nonstationary processes for the distortion noises. We then rigorously characterize the ACF of the wireless channel, using which we provide a comprehensive analysis of four key channel-related metrics: (i) power delay profile (PDP), (ii) coherence time, (iii) coherence bandwidth, and (iv) power spectral density (PSD) of the distortion-plus-noise process. Furthermore, we evaluate these metrics with reasonable UAV wobbling and hardware impairment models to obtain useful insights. Similar to our observation above, this work again demonstrates that the coherence time severely degrades at high frequencies even for small UAV wobbling, which renders air-to-ground channel estimation very difficult at these frequencies. / Doctor of Philosophy / With the increasing maturity of unmanned aerial vehicles (UAVs), also known as drones, wireless ecosystem is changing dramatically. Owing to their ease of deployment and high chance of forming direct line-of-sight (LoS) links with the other UAVs and ground users, they are very appealing for numerous wireless applications. As with any new technology, exploring the full extent of the benefits of UAVs requires careful exploration of the underlying trade space. Therefore, in this dissertation, our main focus is on the analysis of such aerial networks, their interplay with the current terrestrial networks, and the unique features of UAVs that make them different from conventional ground nodes. One important aspect of aerial communication systems is their integration into our current cellular networks. Clearly, the addition of these new aerial components has the potential of benefiting both the ground users (such as mobile users watching a concert who need cellular connectivity to share the moments) and the cellular base station (BS). Therefore, careful analysis of these ``aerial-terrestrial" networks is of utmost importance. In the first phase of this dissertation, we perform this analysis by interpreting the network as a combination of one-hop (from the BS to the user) and two-hop (from the BS to the UAV and then from the UAV to the UE) links. Since the locations of BSs, UAVs, and users are irregular in general, we use tools from stochastic geometry to carry out our analysis, which is a field of mathematics that studies random shapes and patterns. Also, because existing terrestrial BSs are primarily designed to serve the ``ground", we propose the addition of a separate set of antennas at the BS site that is solely used to serve the ``air", i.e., to communicate with the UAVs, and demonstrate the benefits of this additional infrastructure in detail. One of our assumptions in the first phase of this dissertation was that the considered network was static, i.e., the UAVs were hovering in the air and the BSs/users were also not moving. In the second phase, on the other hand, we explore the benefits and challenges of a mobile network of UAVs and characterize the performance of several canonical mobility models in a drone cellular network. In particular, one of the models that we studied extensively is the so-called straight line (SL) mobility model, which was inspired by the simulation models used by the third generation partnership project (3GPP) for the placement and trajectory of UAVs. Since the locations of UAVs could be assumed random in general, we use tools from stochastic geometry and present a unified approach to characterize the point process of UAVs, using which we obtained exact mathematical expressions for the average received rate (i.e., throughput) as seen by the users. Continuing our analysis on mobile setups and using the SL mobility model, we also analyze the handover probability in a drone cellular network, which is defined as the event when the serving UAV of a user changes. By establishing equivalence between our aerial setup with a terrestrial cellular network, we compute the exact handover probability in drone cellular networks. In the final phase of this dissertation, we focus our attention on the air-to-ground wireless channel and attempt to understand its unique features. For that, we propose an impairments-aware unified channel model for an air-to-ground wireless communication system and extensively analyze the link between a hovering UAV in the air and a static user on the ground. In particular, we consider two different types of impairments: (i) UAV wobbling, and (ii) hardware impairments, where the former is caused by random physical fluctuations, and the latter by intrinsic radio frequency (RF) nonidealities at both the transmitter and receiver. Using appropriate models for each type of impairment, we rigorously characterize the autocorrelation function (ACF) of the wireless channel, using which we provide a comprehensive analysis of key channel-related metrics, such as coherence time and coherence bandwidth. One key observation that is revealed from our analysis is that even for small UAV wobbling and low hardware impairment levels, the coherence time of the channel may degrade quickly at high frequencies, which could make it difficult to track the channel and establish a reliable communication link at these frequencies.