Abstract: Although exact symmetries are at best rarely observed experimentally, they play a critically important role in many physical theories. Nonetheless, one suspects theoretical models featuring exact symmetry have some bearing on reality because in various realistic situations the violation of symmetry is, in some intuitive sense, small. In practice, however, it can be very difficult to exploit these ideas unless some sort of a priori division has been made which separates phenomena into a dominant exactly symmetric configuration and small, asymmetric corrections. For example, the notion of an "approximately round" 2-sphere embedded in 3-dimensional space is intuitively clear. But how, given an arbitrary 2-sphere geometry, would one decide whether it is intrinsically "approximately round" or not? This presentation concerns work, still very much in its preliminary stages, which aims to address this and related questions. It will survey a couple open problems in general relativity for which an intrinsic notion of approximate symmetry would be useful, and suggest some basic tools to analyze such situations. These tools include both mathematical definitions of approximate symmetries in general relativity and practical schemes useful to identify those structures in specific space-times. The results, as they currently stand, are mostly negative; it is much easier to rule out ill-conceived approaches that it is to propose plausible solutions. Nonetheless, it appears some fundamental lessons may be drawn even from this preliminary work.