We consider a general notion of \emph{coalgebraic game}, whereby games are viewed as elements of a \emph{final coalgebra}.
This allows for a smooth definition of \emph{game operations} (\emph{e.g.} sum, negation, and linear implication) as \emph{final morphisms}. The notion of coalgebraic game  subsumes different notions of games, \emph{e.g.} possibly non-wellfounded Conway games and games arising in Game Semantics \`a la \cite{AJM00}.
We  define various categories of coalgebraic games and (total) strategies, where the above operations become functorial, and induce a structure of $*$-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 \emph{loopy games}. 

{\bf Keywords}: games,  strategies, categories of games and strategies, Conway games, AJM-games