Interactions in nonlinear elastic media cause multiple scattering and resonances of sound waves, which lead to the loss of phase coherence and acoustic wave degradation through amplitude reduction. However, recently, extraordinary modes of phonon transport have been demonstrated. For instance, granular materials are highly nonlinear prototypical phononic structures that have been extensively studied whereby strong localization of modes such as solitary waves enforce long-range coherent energy propagation [1–6]. High frequency phonons have been shown to propagate over long distance coherently even at room temperature. The occurrence of ballistic transport over distances of many microns, with significant contributions to thermal conductivity [7–9] and coherent propagation through phononic materials with numerous interfaces , have been reported. Consequently, sound-supporting media, phononic structures and acoustic metamaterials offer a broader palette of nonlinear responses with possibility of control of the coherence of phonon propagation. These responses extend over a range of nonlinear types, strengths and orders. These include: (a) geometrical nonlinearity associated with Hertzian contact in granular media [1–6]; (b) intrinsic nonlinearity of the constituent materials and components [11–14]; nonlinear rotational degrees of freedom in composite structures ; (c) hysteretic nonlinearity ; and (d) open system nonlinearity from exchanging matter or energy with an external reservoir . Furthermore, the strength and order of nonlinearity can be selectively amplified in composite media comprising linear and nonlinear constituents . In this Chapter, we present a number of simple models of nonlinear media to illustrate some notions related to nonlinear waves, coherence, and decoherence.