The combined thermocapillary-buoyant flow in sessile and hanging droplets is investigated numerically. The droplet sits on or hangs from a flat plate whose temperature is kept constant. The flow is driven by buoyancy and thermocapillary forces which arise due to a linear variation of the ambient temperature normal to the wall. Neglecting evaporation and in the limit of large mean surface tension the liquid-gas interface is spherical and non-deformable which allows to formulate the problem in body-fitted orthogonal toroidal coordinates such that the interface is a coordinate line. Steady-state axisymmetric solutions to the incompressible Boussinesq equations are obtained using a vorticity-stream function formulation discretized by second-order central finite differences on a non-uniform grid. The resulting nonlinear difference equations are solved iteratively employing a Newton-Raphson method. The results in terms of stream function and temperature are presented varying influential parameters such as the contact angle, Reynolds number, heat transfer rate between the liquid and the ambient, fluid material, and level of gravity. Three different cases are comprehensively examined: thermocapillary-driven flow, buoyancy-driven flow, and mixed thermocapillary-buoyant flow. A temporal two-dimensional linear stability analysis is carried out for the pure buoyant flow as well as for the thermocapillary-driven flow in droplets attached to a flat substrate. The onset of thermal instabilities is found in the buoyancy-driven flow when the temperature is uniformly distributed in vertical direction. Moreover, the existence of axisymmetric instabilities is examined for thermocapillary flow in liquid droplets varying different parameters.