We present an abstract generalization of arc consistency which subsumes the definition of arc consistency in classical CSPs. Our generalization is based on the view of local consistency as technique for approximation of marginal solutions. These approximations are intended for use as heuristics during search. We show that this generalization leads to useful application in classical CSPs as well as non-classical CSPs such as MaxCSP, and instances of the Semi-ring CSP formalism developed by Bistarelli et al. [2]. We demonstrate the application ofthe theory by developing a novel algorithm for use in solving MaxCSP.