What is the computational complexity of "solving" chess?
As @Joe points out, chess is trivial to solve in $O(1)$ time using a lookup table.  (An actual implementation of this algorithm would require a universe significantly larger than the one we live in, but this is a site for theoretical computer science.  The size of the constant is irrelevant.)

There is obviously no canonical $n\times n$ generalization of chess, but several variants have been considered; their complexity depends on how the rules about moves without captures and repeating positions are generalized.

If a draw is declared after a polynomial number of capture-free moves, or after any position repeats a polynomial number of times, then any $n\times n$ chess game ends after a polynomial number of moves, so the problem is clearly in PSPACE.  Storer proved that this variant is PSPACE-hard.

For the variant with no limits on repeated positions or capture-free moves, the number of legal $n\times n$ chess positions is exponential in $n$, so the problem is clearly in EXPTIME.  Fraenkel and Lichtenstein proved that this variant is EXPTIME-hard.