  We present a series of recent results on some new classes of free boundary
problems. Differently from the classical literature, the problems considered
have either a "nonlocal" feature (e.g., the interaction or/and the interfacial
energy may depend on global quantities) or a "nonlinear" flavor (namely, the
total energy is the nonlinear superposition of energy components, thus losing
the standard additivity and scale invariances of the problem).
