Abstract - Complex fluids are characterized by microstructures whose configuration can be driven out of equilibrium by macroscopic flow, and consequently contribute an additional stress that modifies the flow in return. When immiscible complex fluids coexist in a flow system, the fluid-fluid interface introduces an additional length scale into the fluid dynamics, typically intermediate between the microstructures of the bulk fluids and the macroscopic flow. Thus, multicomponent complex fluids often exhibit intriguing features in the interfacial flow, and offer an opportunity to study hydrodynamic coupling across length scales. In this review, we illustrate the novel interfacial dynamics in complex fluids using three examples that involve three types of complex fluids: polymer solutions, liquid crystals and ferrofluids. The focus will be on the dynamics of drops, which coalesce, break up, and self-assemble into regular patterns in these examples. In each case, we present experimental observations of novel interfacial phenomena. Then we explore the underlying fluid-dynamic mechanisms using theoretical models and numerical simulations. Comparing the experimental and computational results, we highlight the roles of the interface in relating complex rheology on the molecular scale to hydrodynamics on the macroscopic scale. Finally we point out outstanding questions and suggest future investigations.