Over the past decade, the significant advancements that have been made in the development of computational tools and numerical techniques have allowed astrophysicists to begin to model accurately the nonlinear growth of gravitational instabilities in a variety of physical systems. The fragmentation or rotationally driven fission of dynamically evolving, self‐gravitating ‘‘drops and bubbles’’ is now routinely modeled in full three‐dimensional generality as we attempt to understand the behavior of protostellar clouds, rotating stars, galaxies, and even the primordial soup that defined the birth of the universe. A brief review is presented here of the general insights that have been gained from studies of this type, followed by a somewhat more detailed description of work, currently underway, that is designed to explain the process of binary star formation. A short video animation sequence, developed in conjunction with some of the research being reviewed, illustrates the basic‐nature of the fission instability in rotating stars and of an instability that can arise in a massive disk that forms in a protostellar cloud.