We extend the improved canonical variational theory, presented previously by us for reaction rates in a classical mechanical world, to reactions in a quantum mechanical world where internal energies of reactants are quantized. We also present a detailed discussion of vibrationally adiabatic models for transmission coefficients for conventional transition-state theory and three versions of variational transition-state theory. We show the relationship of the improved canonical variational transition-state theory to the classical limit of these transmission coefficients. The improved canonical variational theory and the new quantal and classical transmission coefficients are illustrated and tested by applications to quantal collinear and three-dimensional reactions rates for several reactions. We also examine semiclassical approximations to the quantal transmission coefficients. Applications considered are collinear H + H2 and isotopic analogs, Cl + H2 and isotopic analogs, and I + H2 and three-dimensional D + H2, Cl + HD, I + H2, I + D2, O + H2, and F + H2.