We investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values – the “omega one of chess” – denoted by ωCh 1 in the context of finite positions only and by ωCh ～ 1 in the context of all positions, including those with infinitely many pieces. For lower bounds to ωCh ～ 1 , we present specific positions with transfinite game values of ω, ω2, ω2 · k and ω3. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, ωCh ～ 3 1 = ω1.
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