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Phononic materials are periodically arranged building blocks in the form of material properties, geometries, and/or boundary conditions. This synthetic architecture makes phononic materials capable of manipulating mechanical waves that have potential applications across multiple disciplines of physics and engineering. Initial studies have been focused on linear phononic materials that assume small-amplitude waves. The incorporation of nonlinearity, however, has been shown to open opportunities for a new realm of dynamic responses valid beyond the small-amplitude regime. Acknowledging this potential, research in the field has undergone a paradigm shift in the last decade or so by exploiting various sources of nonlinearities within phononic materials. A comprehensive overview of the origin of nonlinearities and how they are modeled, solved, and realized in phononic materials, and specifically, what role nonlinearity plays in enabling rich nonlinear wave responses, is crucial for the future advancement of the field. In this review, we discuss recent advances in nonlinear wave propagation in phononic materials and metamaterials by drawing links between different phononic media and their nonlinearity-induced behaviors. We first briefly discuss the analytical methods employed to solve nonlinear wave propagation problems by focusing on foundational models. We then review physics-based sources of nonlinearities, primarily, material, geometric, and contact nonlinearities and elucidate nonlinear wave responses enabled by them in phononic materials and metamaterials. Finally, we outline existing challenges and possible future directions in nonlinear phononics and metamaterials. [ABSTRACT FROM AUTHOR]