Stretchable electronics has gained global attention due to its potential applications in various fields, including wearable electronics, biomedical engineering, healthcare monitoring, soft robotics, and human-machine interfaces. Stretchable electronics have the unique ability to seamlessly integrate with the constantly deforming human body, known as "biointegration," leading to significant advancements in wearable and implantable biomedical devices. Meanwhile, nanomaterials and nanofabrication technologies have played a crucial role in their development, enabling the integration of electronic materials onto stretchable matrix that can withstand mechanical deformations. Elastomeric materials such as polydimethylsiloxane (PDMS), styrene-butadiene-styrene (SBS), and polyurethane (PU) are commonly used as stretchable substrates, while conducting nanomaterials are incorporated as conductive fillers. These materials address the challenges posed by the mechanical mismatch between traditional rigid electronics and biological tissues, offering improved flexibility, enhanced device-tissue interface, and reduced immunological reactions. By incorporating appropriate fillers into elastomers, researchers aim to develop intrinsically soft bio-integrated electronics that monitor electrophysiological as well as physical signals coming from biological tissues without hindrance from mechanical mismatch between electronics and soft tissues. This comprehensive review explores the materials and functionalization techniques of elastomers, as well as the development of functionalized elastomer-based bio-integrated devices such as sensors and stimulators. Future research directions and challenges in the field of stretchable electronics and bio-integrated devices are also discussed.