The paper reviews the development and application of diffuse-interface models in fluid mechanics, particularly for interfacial phenomena. These models are useful when the length scale of the interface is comparable to or smaller than the length scale of the phenomena being studied. The authors discuss the formulation of diffuse-interface models for single-component and binary fluids, including equilibrium and non-equilibrium situations. They also review recent applications and computations using these models, such as near-critical fluid dynamics, droplet breakup and coalescence, and the development of hydrodynamic instabilities. The paper addresses issues including sharp-interface analyses, computational methods for describing interfacial phenomena, and models of fully miscible fluids. The authors highlight the advantages of diffuse-interface models over classical free-boundary formulations, particularly in situations where the interface thickness is significant. They provide detailed derivations and numerical examples to illustrate the effectiveness of these models in various physical scenarios.The paper reviews the development and application of diffuse-interface models in fluid mechanics, particularly for interfacial phenomena. These models are useful when the length scale of the interface is comparable to or smaller than the length scale of the phenomena being studied. The authors discuss the formulation of diffuse-interface models for single-component and binary fluids, including equilibrium and non-equilibrium situations. They also review recent applications and computations using these models, such as near-critical fluid dynamics, droplet breakup and coalescence, and the development of hydrodynamic instabilities. The paper addresses issues including sharp-interface analyses, computational methods for describing interfacial phenomena, and models of fully miscible fluids. The authors highlight the advantages of diffuse-interface models over classical free-boundary formulations, particularly in situations where the interface thickness is significant. They provide detailed derivations and numerical examples to illustrate the effectiveness of these models in various physical scenarios.