  We review some recent results on the equilibrium shapes of charged liquid
drops. We show that the natural variational model is ill-posed and how this can
be overcome by either restricting the class of competitors or by adding
penalizations in the functional. The original contribution of this note is
twofold. First, we prove existence of an optimal distribution of charge for a
conducting drop subject to an external electric field. Second, we prove that
there exists no optimal conducting drop in this setting.
