Development and utilization of ionic liquids for applications as electrolytes and solvents requires understanding their electrostatic interactions with solutes, co-solvents, and interfaces. In this regard, the bulk dielectric constant of an ionic liquid is of little use, but rather the dielectric response must be characterized at atomistic lengthscales. We will discuss the dielectric response of ionic liquids as computed from molecular dynamics simulations and statistical mechanical theory. We will then describe how the dielectric properties of ionic liquids are connected with observed trends in interfacial capacitance, solvation, and liquid structure. Comparisons will be made to aqueous electrolytes and molten salts, and similarities and differences will be discussed.