We study quantum cosmology with conformal matter comprising a perfect radiation fluid and a number of conformally coupled scalar fields. Focusing initially on the collective coordinates (minisuperspace) associated with homogeneous, isotropic backgrounds, we are able to perform the quantum gravity path integral exactly. The evolution describes a "perfect bounce", in which the Universe passes smoothly through the singularity. We extend the analysis to spatially flat, anisotropic universes, treated exactly, and to generic inhomogeneous, anisotropic perturbations treated at linear and nonlinear order. This picture provides a natural, unitary description of quantum mechanical evolution across a cosmological bounce. We provide evidence for a semiclassical description in which all fields pass "around" the cosmological singularity along complex classical paths.