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It is the aim of this paper to generalize existing connections between automata and logic to a more general, coalgebraic level. Let F:Set-Set be a standard functor that preserves weak pullbacks. We introduce the notion of an F-automaton, a device that operates on pointed F-coalgebras; the criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an F-automaton A"p which is equivalent to p in the sense that A"p accepts precisely those pointed F-coalgebras in which p holds.