The stability with respect to two- and three-dimensional perturbations of natural-convection flow of air in a square enclosure with differentially heated vertical walls and periodic boundary conditions in the lateral direction has been investigated. The horizontal walls are either conducting or adiabatic. The solution is numerically approximated by Chebyshev–Fourier expansions. In contrast to the assumption made in earlier studies, three-dimensional perturbations turn out to be less stable than two-dimensional perturbations, giving a lower critical Rayleigh number in the three-dimensional case for the onset of transition to turbulence. Both the line-symmetric and line-skew-symmetric three-dimensional perturbations are found to be unstable. The most unstable wavelengths in the lateral direction typically are of the same size as the enclosure. In the nonlinear solution new symmetry breaking occurs, giving either a steady or an oscillating final state. The three-dimensional structures in the nonlinear saturated solution consist of counter-rotating longitudinal convection rolls along the horizontal walls. The energy balance shows that the three-dimensional instabilities have a combined thermal and hydrodynamic nature. Besides the stability calculations, two- and three-dimensional direct numerical simulations of the weakly turbulent flow were performed for the square conducting enclosure at the Rayleigh number 108. In the two-dimensional case, the time-dependent temperature shows different dominant frequencies in the horizontal boundary layers, vertical boundary layers and core region, respectively. In the three-dimensional case almost the same frequencies are found, except for the horizontal boundary layers. The strong three-dimensional mixing leaves no, or only very weak, three-dimensional structures in the time-averaged nonlinear solution. Three-dimensional effects increase the maximum of the time- and depth-averaged wall-heat transfer by 15%.