Electronic products enhance our lives and bring change in almost every field including communications, entertainment, and healthcare. However, the fast-growing surplus of electronic waste has become a serious challenge to the environment: the majority of these toxic materials are deposited directly into landfills or incinerated because recycling is difficult and/or expensive. Electronic devices manufactured with biocompatible and biodegradable polymers are in demand because they are environmentally safe, of low cost, of potentially large volume, and are disposable. In addition, biocompatible and biodegradable electronics may enable the use of electronic devices in implantable biomedical applications. This review is focused on polymer materials, both natural and synthetic, that are utilized in biodegradable and/or biocompatible electronic devices as substrates, dielectrics, conductors, or semiconductors. Specific application examples are highlighted, such as biodegradable electronic devices, implantable medical devices, and edible sensors.