Abstract: The liquid drop model, an old problem introduced by Gamow for the shape of atomic nuclei and nuclear fission, has recently resurfaced within the framework of the modern calculus of variations. Namely, this model is given by a constrained nonconvex nonlocal geometric variational problem. In this talk, I will present the current state of the art for the existence and nonexistence of minimizers of the liquid drop problem. I will also focus on this model on a finite domain, and describe its relation to the Ohta-Kawasaki theory for self-assembly of diblock copolymers. Finally, I will discuss my recent work on a regularization of the liquid drop model which enables us to describe minimizing sequences quantitatively and understand the droplet breakup in the nonexistence regime.