The relentless demand for data in our society has driven the continuous evolution of wireless technologies to enhance network capacity. While current deployments of 5G have made strides in this direction using massive multiple-input-multiple-output (MIMO) and millimeter-wave (mmWave) bands, all existing wireless systems operate in a half-duplex (HD) mode. Full-duplex (FD) wireless communication, on the other hand, enables simultaneous transmission and reception (STAR) of signals at the same frequency, offering advantages such as enhanced spectrum efficiency, improved data rates, and reduced latency. This article presents a comprehensive review of FD wireless systems, with a focus on hardware design, implementation, cross-layered considerations, and applications. The major bottleneck in achieving FD communication is the presence of self-interference (SI) signals from the transmitter (TX) to the receiver, and achieving SI cancellation (SIC) with real-time adaption is critical for FD deployment. The review starts by establishing a system-level understanding of FD wireless systems, followed by a review of the architectures of antenna interfaces and integrated RF and baseband (BB) SI cancellers, which show promise in enabling low-cost, small-form-factor, portable FD systems. We then discuss digital cancellation techniques, including digital signal processing (DSP)- and learning-based algorithms. The challenges presented by FD phased-array and MIMO systems are discussed, followed by system-level aspects, including optimization algorithms, opportunities in the higher layers of the networking protocol stack, and testbed integration. Finally, the relevance of FD systems in applications such as next-generation (xG) wireless, mmWave repeaters, radars, and noncommunication domains is highlighted. Overall, this comprehensive review provides valuable insights into the design, implementation, and applications of FD wireless systems while opening up new directions for future research.