The goal of this thesis is to study of the evolution of 3D electrically charged liquid droplets of fluid evolving under the influence of surface tension and electrostatic forces. In the first part of the thesis, an appropriate mathematical model of the problem is introduced and the linear stability analysis is developed by perturbing a sphere with spherical harmonics. In the second part, the numerical solution of the problem is described with the use of the boundary elements method (BEM) on an adaptive mesh of triangular elements. The numerical method is validated by comparison with exact solutions. Finally, various numerical results are presented. These include neck formation in droplets, the evolution of surfaces with holes, singularity formation on droplets with various symmetries and numerical evidence that oblate spheroids are unstable.