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We consider a general notion of coalgebraic game, whereby games are viewed as elements of a final coalgebra. This allows for a smooth definition of game operations (e.g. sum, negation, and linear implication) as final morphisms. The notion of coalgebraic game subsumes different notions of games, e.g. possibly non-wellfounded Conway games and games arising in Game Semantics à la [AJM00]. We define various categories of coalgebraic games and (total) strategies, where the above operations become functorial, and induce a structure of monoidal closed or *-autonomous category. In particular, we define a category of coalgebraic games corresponding to AJM-games and winning strategies, and a generalization to non-wellfounded games of Joyal's category of Conway games. This latter construction provides a categorical characterization of the equivalence by Berlekamp, Conway, Guy on loopy games.