A numerical study of steady-state and transient 3-D Marangoni convection and heat transfer in electrostatically levitated droplets. The analysis is based on the Galerkin finite element solution of the Navier-Stokes and energy equations. Numerical aspects for the computation of surface driven flows in general curvilinear coordinate system are discussed within the framework of finite elements and differential geometry. Results show that for practical micro-gravity conditions under which the deformation is small, the single and double beam heating arrangements, when placed at the poles or equator plane, produce an approximately axisymmetric flow profile and temperature distribution with the axis of symmetry defined by the line passing through the centers of the laser beam and the droplet. Thus, an axisymmetric model could provide a reasonably good description, and an exact one when the laser beam is placed at a pole or one beam is placed at both poles. When a tetrahedral or octahedral heating arrangement is applied, complex 3-D flow structures occur, which result from interaction of flow motions asociated with each laser beam. For the case studied, the tetrahedral heating arrangement does not seem to produce a significant reduction in internal velocity, in contrast to perception, but the temperature is more uniform. This phenomenon is explained by the fact that the 2- and 4-beam arrangements result in a surface temperature gradient of approximately the same magnitude. The six-beam heating placement produces, however, a much more significant reduction in both velocity and temperature non-uniformity. The transient decaying during cooling is characterized by the evolution of both temperature and velocity fields evolve in a rather complex fashion, with the initial stage dominated by the pronounced thermal and flow mixing on the surface layer of a droplet. The strong surface mixing quickly brings out a surface temperature distribution of axi-symmetry, while a 3-D structure still prevails inside until much later. The flow reversal is also observed in the droplets that have been heated by 4- or 6-beam lasers during the decay, and there is a spike in velocity and temperature at the time when the flow reversal occurs. (C) 2004 Elsevier Ltd. All rights reserved.