  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---with two senses depending on whether one considers only
finite positions or also positions with infinitely many pieces. For lower
bounds, we present specific infinite positions with transfinite game values of
omega, omega^2, omega^2 times k, and omega^3. 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, true omega one.
