A computational study of the character and stability of two-dimensional buoyant thermocapillary flows, valid to leading order in capillary number (Ca), is conducted in the Grashof number (Gr), Reynolds number (Re), aspect ratio, and Prandtl number (Pr) parameter space. Calculations of thermocapillary convection for low Pr fluids have generally produced steady results. Calculations of pure buoyant convection (Re=0) exhibit a Hopf bifurcation at Grcr (no thermocapillarity) that is well understood. Thus, the combined thermocapillary buoyant problem is studied to investigate the onset of oscillatory convection in the limit Gr→0. The unsteady natural convection pattern at fixed Gr≳Grcr is modified only slightly for low values of Re. When thermocapillarity acts in conjunction with buoyancy (Re≳0) it is stabilizing, in that the transition to unsteady flow occurs at Gr≳Grcr, as defined for the strictly buoyant problem. When thermocapillarity acts in opposition to buoyancy (Re<0), it is destabilizing for relatively small values of ‖Re‖, but thermocapillarity ultimately dominates the convective pattern for larger ‖Re‖, and the resulting flow is steady for the range of parameter values considered. Stability boundaries for the onset of oscillatory convection in the Gr–Re plane are given for representative values of the cavity aspect ratio and Pr.