When the free surfaces of most solids approach their bulk melting temperatures from below, the molecular structure of the material gives way to a disordered structure with some attributes of both the solid and liquid phases. When the temperature is sufficiently close to that of bulk transition, the surface melts and literally flows as a viscous fluid. This phenomenon, called interfacial premelting, lies at the heart of the microscopic theory of melting of solid matter, and captures the interest of condensed matter physicists and physical chemists alike. The process is ubiquitous and responsible for a wide range of consequences in materials with biological, geophysical, and technological significance. Because such systems are often exposed to spatial or temporal variations in thermodynamic forcing, there are a host of fluid mechanical phenomena that result from this underlying melting behavior. The fluid dynamics of unfrozen surfaces holds clues for understanding the bulk behavior of polycrystalline materials, from Earth’s mantle to the stratosphere and beyond. In this review we focus on the fluid dynamical consequences of the premelting of solids.