A superconductor is a perfect conductor and a perfect diamagnet in which currents flow with zero resistance and screen external magnetic fields. This emergent state of electrons has been described successfully by the Bardeen-Cooper-Schrieffer (BCS) paradigm. Within this standard paradigm, superconductivity arises when the Fermi surface in a normal metal becomes unstable to pairing. What happens when this paradigm breaks down, for example, in systems with very small Fermi surfaces, or in the absence of a Fermi surface altogether, as in a band insulator? In this talk I will provide some answers and also show how exciting new directions for superconductivity and superfluidity research are opening up in quantum materials and ultra-cold atomic gases.