The dynamics of mobile inclusions in lipid membranes are fundamental to a variety of biological processes, including the aggregation of proteins during cell-cell signaling and the coalescence of lipid raft domains. Additionally, elucidating the mobilities and hydrodynamic interactions of colloidal particles at a fluid-fluid interface has important technological ramifications on the formation of colloidosomes. In many of these applications, the membrane/interface is a compact, strongly curved structure. In this talk, I will explore the effects of curvature and topology on the hydrodynamics and transport properties of membranes. In particular, I will consider the motion of point-like and extended (rod-like) objects in a spherical membrane. The results indicate that the topology and curvature of a membrane can indeed have a strong effect on its transport properties: First, the topology of the sphere fundamentally alters the nature of the membrane velocity field by forcing the existence of singularities (e.g. vortices, sources, sinks) in this field; in addition, the curvature of the membrane can suppress the diffusion of objects embedded therein. Finally, I will show that this theoretical model agrees quantitatively with recent experimental measurements by A.D. Dinsmore and co-workers (University of Massachusetts, Amherst) on the motion of a rod bound to the crowded interface of a water-in-oil droplet.