Turbulent natural convection with and without radiation transfer in two-dimensional (2D) and three-dimensional (3D) air-filled differentially heated cavities is numerically investigated using various RANS (Reynolds Averaged Navier-Stokes) turbulence models and the Discrete Ordinates radiation model. Five different two-equation eddy-viscosity models including the standard k-epsilon model, the renormalization group (RNG) k-epsilon model, the realisable k-epsilon model, the standard k-omega model and the shear-stress transport (SST) k-omega model are selected for comparison. Qualitative and quantitative data are presented to demonstrate the effects of three-dimensionality, radiation transfer and the thermal boundary conditions on the horizontal surfaces on the numerical solution of the convective flow in the cavity. The present numerical results are compared against published experimental and direct numerical simulation data. It is found that the predicted thermal stratification in the interior of the cavity is improved when the simulation is extended from 2D to 3D and when the effect of radiation transfer is accounted for. The discrepancy with regard to the interior stratification between the experiment and numerical simulation is mainly caused by the negligence of radiation transfer. The thermal boundary conditions on the horizontal surfaces also have a significant impact on the numerical solution, especially when the radiation transfer is not accounted for. Further, the present results show that all the RANS models are capable of capturing the main features of the flow and the overall performance of these turbulence models in terms of predicting time-averaged quantities is acceptable. It is found that the variation of the numerical results obtained with the three k-epsilon models is very small, whereas the discrepancy between the two k-omega models is significant. The SST k-omega model has the best overall performance and the standard k-omega model has the worst overall performance. (C) 2015 Elsevier Ltd. All rights reserved.