Thermocapillary convection appears in many polymer processing operations. In order to understand the effect of gravity for its stability, three-dimensional linear stability analysis is performed for a viscoelastic fluid. The critical Marangoni number is derived as a function of Bond number, Prandtl number and elasticity. When the Prandtl number is large, the increasing of gravity effect often makes the flow more stable, and the coupling of gravity to elasticity and thermocapillary force excites many different kinds of preferred modes. For small Prandtl number, the flow is always destabilized by gravity effect. The work done by gravity becomes a new energy source for perturbation. However, it can be either positive or negative, which is not directly related to the variation of critical Marangoni number with gravity. The effect of gravity on the instability mechanism and the properties of preferred modes are demonstrated for different elasticity. (C) 2018 Elsevier Ltd. All rights reserved.