  We do not yet know how to quantize gravity in 3+1 dimensions, but in lower
dimensions we face the opposite problem: many of the approaches originally
developed for (3+1)-dimensional gravity can be successfully implemented in 2+1
dimensions, but the resulting quantum theories are not all equivalent. In this
talk, I discuss six such approaches --- reduced phase space ADM quantization,
the Chern-Simons/connection representation, covariant canonical quantization,
the loop representation, the Wheeler-DeWitt equation, and lattice methods ---
for the simple case of a spacetime whose spatial topology is that of a torus. A
comparison of the resulting quantum theories can provide some useful insights
into the conceptual issues that underlie quantum gravity in any number of
dimensions. (Talk given at the Fifth Canadian Conference on General Relativity
and Relativistic Astrophysics, Waterloo, Ontario, May 1993)
