Despite the great importance of heterogeneous catalysis, research in this field has long been characterized by its empiricism. Now, however, thanks to the rapid development of methods in surface physics, the elementary steps can be identified at the atomic level and the underlying principles understood. Defined single crystal surfaces are employed as models, based on the analysis of the surfaces of ‘real’ catalysts. Direct images, with atomic resolution, can be obtained using scanning tunneling microscopy, while electron spectroscopic methods yield detailed information on the bonding state of adsorbed species and the influence of catalyst additives (promotors) upon them. The successful application of this approach is illustrated with reference to the elucidation of the mechanism of ammonia synthesis. The catalyst surface is usually transformed under reaction conditions, and, as the processes involved are far-removed from equilibrium, such transformations can lead to intrinsic spatial and temporal self-organization phenomena. In this case, the reaction rate may not remain constant under otherwise invariant conditions but will change periodically or exhibit chaotic behavior, with the formation of spatial patterns on the catalyst surface.