Packings of soft frictionless spheres are a widely used model for emulsions and even granulates. Their study has taken a life of its own because the structures that these spheres make when you press them together ever so gently are mechanically very peculiar. They are very weird solids which resist compression much more than they resist shear, which only behave elastically on very large length scales. It is safe to say that their study introduces many physicists to concepts in rigidity theory and structural mechanics that were before only well-known to mathematicians and engineers, and this has led to many interesting new directions of research.