Transparent electrodes that can remain electrically conductive and stable under large mechanical deformations are highly desirable for applications in flexible and wearable electronics. This paper describes a comprehensive study of the electrical, optical, and mechanical properties of hybrid nanostructures based on two-dimensional graphene and networks of one-dimensional metal nanowires, and their use as transparent and stretchable electrodes. Low sheet resistance (33 Ω/sq) with high transmittance (94% in visible range), robust stability against electric breakdown and oxidation, and superb flexibility (27% in bending strain) and stretchability (100% in tensile strain) are observed, and these multiple functionalities of the hybrid structures suggest a future promise for next generation electronics. The use of hybrid electrodes to fabricate oxide semiconductor transistors and single-pixel displays integrated on wearable soft contact lenses with in vivo tests are demonstrated.