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In this article, we review the key concepts and the progress in the design of physical-layer aspects of in-band full-duplex (IBFD) communications. One of the fundamental challenges in realizing IBFD is self-interference that can be up to 100 dB stronger than signals of interest. Thus, we start by reviewing state-of-the-art eesearch in self-interference cancellation, addressing both model-based and emerging machine learning-based methods. Then, we turn our attention to new wireless systems with many degrees of freedom for which the traditional IBFD designs do not gracefully scale and, hence, require many innovations to enable IBFD. We provide an extensive review of basic concepts and state of the art in massive multiple-input–multiple-output IBFD. Then, we consider the mmWave band IBFD and review advanced physical-layer architectures. The above review provides the proper context to discuss IBFD innovations and new challenges for sixth-generation networks and beyond, where wireless networks are envisioned to be multifunctional, combining communications with functions such as sensing, cognitive radios, physical-layer security, and wireless power transfer. We conclude this article with a status update on the adoption of IBFD in communication standards.