The need for transparent conductors (TCs) that are capable of withstanding high mechanical deformation in comparison to the brittle indium tin oxide (ITO) films is paramount for roll-to-roll production of flexible and stretchable displays, signage systems, lighting devices and solar panels with stringent weatherability requirements. This paper reports a highly stretchable TC comprising of a web of core-shell nanofibers, which mimics the fibrous structure of natural systems such as veins of a leaf or nerve systems. The TC web demonstrates high transparency, low sheet resistance, and unprecedented stretchability and stability over repeated stretching. The nanofiber TC web can be transferred to different substrates, which is manifested by the transfer onto an organic solar cell, demonstrating a photovoltaic performance comparable to that of a device with an ITO electrode. This work presents a technological platform, scalable for the manufacturing of large area transparent conductors for flexible and stretchable displays, electronics and solar cells on unconventional substrates such as rubber, fabric and paper.