Benchmark numerical solutions for a three-dimensional natural convection heat transfer problem in a cubical cavity are presented in this paper. The 3-D cavity has two differentially heated and isothermal vertical walls and also four adiabatic walls. The computations are conducted for three Rayleigh numbers of 10(4), 10(5) and 10(6). The filled fluid is with air and the Prandtl number is fixed at 0.71. The computed results are efficiently obtained by using the time-space method, which was proposed by Saitoh (199 1) as a highly efficient and fast solver for general heat transfer and fluid flow problems. In our computations, the high-accuracy finite differences of a fourth-order were employed for the spatial discretization of governing equations and boundary conditions. In addition the third-order backward finite difference was used in timewise discretization. The resultant converged flow and temperature characteristics are also presented. The spatial grid dependency of the solutions was examined on a uniform grid. In addition, the grid-independent benchmark solutions were obtained by Richardson extrapolation for three cases. The present benchmark solutions will be useful for checking the performance and accuracy of any numerical methodologies. (C) 2003 Elsevier Ltd. All rights reserved.