Persistent currents are a hallmark of both superfluidity and superconductivity. Just as a current in a superconducting circuit will flow forever, if a current is created in a superfluid Bose-Einstein condensate, the flow will not decay, as long as the current is below a critical value. Ultracold atomic gases are an ideal system for studying superfluid behavior. By manipulating optical potentials, we are able to create ring-shaped Bose-Einstein condensates. Recently, we have used this "atom circuit" to create and study long-lived persistent currents. The persistent currents can be controlled with the addition of an additional laser beam, which acts as a barrier or "weak link" across one side of the ring. Weak connections between superconductors or superfluids can differ from classical links due to quantum coherence. By tuning the properties of our weak link we are able to drive transitions between persistent current states, and we have studied how those transitions are affected by the strength of the perturbation and the temperature of the system. The properties of a weak link are characterized by a single function, the current-phase relationship. In recent experiments, we have developed a technique to directly measure the current-phase relationship of the weak link, and have demonstrated that the system also exhibits hysteresis. In electronic circuits, hysteresis plays an important role, particularly in applications like memory and digital noise filters. It’s possible in future "atomtronic" circuits, our device could play a similar role.