Interest in new thermal diodes, regulators, and switches has been rapidly growing because these components have the potential for rich transport phenomena that cannot be achieved using traditional thermal resistors and capacitors. Each of these thermal components has a signature functionality: Thermal diodes can rectify heat currents, thermal regulators can maintain a desired temperature, and thermal switches can actively control the heat transfer. Here, we review the fundamental physical mechanisms of switchable and nonlinear heat transfer which have been harnessed to make thermal diodes, switches, and regulators. The review focuses on experimental demonstrations, mainly near room temperature, and spans the fields of heat conduction, convection, and radiation. We emphasize the changes in thermal properties across phase transitions and thermal switching using electric and magnetic fields. After surveying fundamental mechanisms, we present various nonlinear and active thermal circuits that are based on analogies with well-known electrical circuits, and analyze potential applications in solid-state refrigeration and waste heat scavenging.