With the continuous increase in the production of electronic devices, large amounts of electronic waste (E-waste) are routinely being discarded into the environment. This causes serious environmental and ecological problems because of the non-degradable polymers, released hazardous chemicals, and toxic heavy metals. The appearance of biodegradable polymers, which can be degraded or dissolved into the surrounding environment with no pollution, is promising for effectively relieving the environmental burden. Additionally, biodegradable polymers are usually biocompatible, which enables electronics to be used in implantable biomedical applications. However, for some specific application requirements, such as flexibility, electric conductivity, dielectric property, gas and water vapor barrier, most biodegradable polymers are inadequate. Recent research has focused on the preparation of nanocomposites by incorporating nanofillers into biopolymers, so as to endow them with functional characteristics, while simultaneously maintaining effective biodegradability and biocompatibility. As such, bionanocomposites have broad application prospects in electronic devices. In this paper, emergent biodegradable and biocompatible polymers used as insulators or (semi)conductors are first reviewed, followed by biodegradable and biocompatible nanocomposites applied in electronics as substrates, (semi)conductors and dielectrics, as well as electronic packaging, which is highlighted with specific examples. To finish, future directions of the biodegradable and biocompatible nanocomposites, as well as the challenges, that must be overcome are discussed.