Using \emph{coalgebraic methods}, we extend Conway's  theory of games to  possibly  \emph{non-terminating}, 
\emph{i.e.} \emph{non-wellfounded games}  (\emph{hypergames}).
We take the view that a play which goes on forever is a \emph{draw}, and hence rather than focussing on 
winning strategies, we focus on \emph{non-losing strategies}.
Hypergames are a fruitful metaphor for non-terminating processes, \emph{Conway's sum}  being similar to 
\emph{shuffling}.
We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we 
generalize Conway's results  on game \emph{determinacy} and \emph{characterization} of strategies.
Hypergames have a  rather interesting theory, already in the case of
\emph{impartial hypergames}, for which we give a \emph{compositional semantics}, in terms of  a 
\emph{generalized Grundy-Sprague function} and a system of generalized \emph{Nim games}. 
\emph{Equivalences} and \emph{congruences} on games and hypergames are discussed.
We indicate a number of intriguing directions for future work.
We briefly compare hypergames with other notions of games used in computer science.