I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. In the talk, I will focus on Rayleigh's model of perfectly conducting liquid drops and discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models.