We survey recent results on logics with counting and their local properties. We first consider game-theoretic characterizations of first-order logic and its counting extensions provided by unary generalized quantifiers. We then study Gaifman's and Hanf's locality theorems, their connection with game characterizations, and examples of their usage in proving expressivity bounds for first-order logic and its extensions. We review the abstract notions of Gaifman's and Hanf's locality, and show how they are related. We also consider a closely related bounded degree property, and demonstrate its usefulness in proving expressivity bounds. We discuss two applications in computer science. One deals with proving lower bounds for the class TC 0 . In particular, we use logical characterization of TC 0 and locality theorems for first-order with counting quantifiers to provide lower bounds. We then explain how the notions of locality are used in database theory to prove that extension..
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