This paper examines a class of linear-quadratic discrete-time games. Attention is paid to the problem of ascertaining the existence of the open-loop and closed-loop solutions as well as to the forms of these solutions for games with a finite horizon. The new contributions of the paper to existing results for this class of games cover the following points : the introduction of the concepts of nonunique optimal strategies, of the pair of ‘ minimum energy ’ optimal strategies, the characterization of the so-called ‘no-conjugate-point condition’ in these discrete-time games. These results also have various implications, showing that this class of games has peculiar features not shared by its continuous-time counterpart. Finally, an attempt is made to extend the results to the case of games with infinite horizon. Only a few explicit results are obtained in this direction, with the main difficulties indicated.