Recent advances in chemical vapour deposition have led to the fabrication of large graphene sheets on metal foils for use in research and development. However, further breakthroughs are required in the way these graphenes are transferred from their growth substrates onto the final substrate. Although various methods have been developed, as yet there is no general way to reliably transfer graphene onto arbitrary surfaces, such as 'soft' ones. Here, we report a method that allows the graphene to be transferred with high fidelity at the desired location on almost all surfaces, including fragile polymer thin films and hydrophobic surfaces. The method relies on a sacrificial 'self-releasing' polymer layer placed between a conventional polydimethylsiloxane elastomer stamp and the graphene that is to be transferred. This self-releasing layer provides a low work of adhesion on the stamp, which facilitates delamination of the graphene and its placement on the new substrate. To demonstrate the generality and reliability of our method, we fabricate high field-strength polymer capacitors using graphene as the top contact over a polymer dielectric thin film. These capacitors show superior dielectric breakdown characteristics compared with those made with evaporated metal top contacts. Furthermore, we fabricate low-operation-voltage organic field-effect transistors using graphene as the gate electrode placed over a thin polymer gate dielectric layer. We finally demonstrate an artificial graphite intercalation compound by stacking alternate monolayers of graphene and 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F(4)TCNQ). This compound, which comprises graphene sheets p-doped by partial hole transfer from the F(4)TCNQ, shows a high and remarkably stable hole conductivity, even when heated in the presence of moisture.