In nature, bioelectricity refers to electrical potentials and currents produced by or used within living cells, tissues, and organisms for electrolocation, predation, or protection. Electric eels can generate huge amounts of power using electric organs that are arranged in stacks of electrocytes. One critical, recent issue for electronic gadgets is that energy storage systems are incapable of offering enough energy for uninterrupted, long-running processes. This results in frequent recharging or inconvenient energy storage unit replacement. To address this challenge, inspired by bioelectricity phenomena, a unique triboelectric nanogenerator (TENG) technology has been proposed to convert small quantities of mechanical energy into electricity without an external power supply. Several advances have been made regarding TENG-based self-charging energy storage devices. To fulfill the sustainable operation requirements of the next-generation electronic devices, extensive work has been performed to integrate energy-generating TENGs with energy-storing supercapacitor devices to form self-charging power systems (SCPSs). This tutorial article focuses on the recent advances and various SCPS structural designs. In addition, various power management circuits that can be integrated with TENG devices and supercapacitors are reviewed. Finally, challenges and perspectives for future SCPS progress are discussed.