Steady-state shear viscosity eta((gamma) over dot) of unconcatenated ring polymer melts as a function of the shear rate (gamma) over dot is studied by a combination of experiments, simulations, and theory. Experiments using polystyrenes with Z approximate to 5 and Z approximate to 11 entanglements indicate weaker shear thinning for rings compared to linear polymers exhibiting power law scaling of shear viscosity eta similar to (gamma) over dot(-0.56 +/- 0.02), independent of chain length, for Weissenberg numbers up to about 10(2). Nonequilibrium molecular dynamics simulations using the bead-spring model reveal a similar behavior with eta similar to (gamma) over dot(-0.57 +/- 0.08) for 4 <= Z <= 57. Viscosity decreases with chain length for high (gamma) over dot. In our experiments, we see the onset of this regime, and in simulations, which we extended to Wi similar to 10(4), the nonuniversality is fully developed. In addition to a naive scaling theory yielding for the universal regime eta similar to (gamma) over dot(-0.57), we developed a novel shear slit model explaining many details of observed conformations and dynamics as well as the chain length-dependent behavior of viscosity at large (gamma) over dot The signature feature of the model is the presence of two distinct length scales: the size of tension blobs and much larger thickness of a shear slit in which rings are self-consistently confined in the velocity gradient direction and which is dictated by the size of a chain section with relaxation time 1/(gamma) over dot These two length scales control the two normal stress differences. In this model, the chain length-dependent onset of nonuniversal behavior is set by tension blobs becoming as small as about one Kuhn segment. This model explains the approximate applicability of the Cox-Merz rule for ring polymers.