The process of diffusion is an intriguing one, both from a conceptual and from a practical point of view. From a conceptual standpoint a striking number of fundamental theoretical developments owe their discovery to an analysis of diffusion. For example, Brownian motion (1, 2) played a central role in the development of atomic theory by providing one of the more direct bridges between the microscopic and macroscopic worlds. Likewise, the analysis of mathematical techniques aimed at solving diffusion equations led, in part, to the development of a number of modern statistical methods (3, 4), methods that are proving extremely useful in the analysis of a wide variety of classical (5) and quantum-mechanical (6-8) problems. More recently, a careful analysis (9, 10) of the time decay of velocity autocorrelation functions that describe diffusion led to the discovery of long time tails in such functions and contributed greatly to our general understanding of transport phenomena. From a practical point of view, diffusion is an important, often crucial, elementary con densed-phase process. For example, it frequently provides the basic path way by which interfacial reactants are brought into physical proximity, thereby setting the natural time scale for such chemical processes. The present review focuses attention specifically on the topic of surface diffusion. For reasons of space limitations we further restrict the discussion to the diffusion of single adsorbates and two-dimensional clusters, noting that some of the methods we discuss can be trivially extended to treat