Variational problems stemming from continuum physics continue to drive many scientific communities from mathematicians working in the modern calculus of variations to material scientists. These problems, often nonconvex and nonlocal, are simple to state yet have rich and complex structure. In this talk I will survey three such problems: (i) Gamow's Liquid Drop problem, originally proposed for the shape of atomic nuclei; (ii) A nonlocal interaction problem stemming from the modelling of swarms; (iii) The phase field crystal problem for grain morphology in polycrystalline materials. In each case, I will address their alluring simplicity and complexity, and some of the pure and applied mathematics which surround foundational questions.