We describe the design and application of a low-energy electron microscope (LEEM) dedicated to the study of III-V materials. Recent studies of Langmuir (free) evaporation of GaAs(001) have been reviewed. Running Ga droplets are observed, and the motion is predicted and shown to slow and stop near a characteristic temperature. Striking bursts of daughter droplet nucleation accompany the coalescence of large parent droplets. These observations imply that evaporation and surface morphology are intimately connected, suggesting a new approach for the self-assembly and positioning of nanostructures on patterned surfaces.