Reconfigurable intelligent surfaces (RISs) are considered to be a key enabler of highly energy-efficient 6G and beyond networks. This property arises from the absence of power amplifiers in the structure, in contrast to active nodes, such as small cells and relays. However, still an amount of power is required for their operation. To improve their energy efficiency further, we propose the notion of perpetual RISs, which secure the power needed to supply their functionalities through wireless energy harvesting of the impinging transmitted electromagnetic signals. Towards this, we initially explain the rationale behind such RIS capability and proceed with the presentation of the main RIS controller architecture that can realize this vision under an in-band energy harvesting consideration. Furthermore, we present a typical energy-harvesting architecture followed by two harvesting protocols. Subsequently, we study the performance of the two protocols under a typical communications scenario. Finally, we elaborate on the main research challenges governing the realization of large-scale networks with perpetual RISs.