Reliable transfer techniques are critical for the integration of 2D materials with arbitrary substrates. Here, the authors describe a method to transfer 4-inch and A4-sized defect-free graphene films onto rigid and flexible substrates with controllable conformal contact, leading to improved electrical properties and uniformity. The availability of graphene and other two-dimensional (2D) materials on a wide range of substrates forms the basis for large-area applications, such as graphene integration with silicon-based technologies, which requires graphene on silicon with outperforming carrier mobilities. However, 2D materials were only produced on limited archetypal substrates by chemical vapor deposition approaches. Reliable after-growth transfer techniques, that do not produce cracks, contamination, and wrinkles, are critical for layering 2D materials onto arbitrary substrates. Here we show that, by incorporating oxhydryl groups-containing volatile molecules, the supporting films can be deformed under heat to achieve a controllable conformal contact, enabling the large-area transfer of 2D films without cracks, contamination, and wrinkles. The resulting conformity with enhanced adhesion facilitates the direct delamination of supporting films from graphene, providing ultraclean surfaces and carrier mobilities up to 1,420,000 cm(2) V-1 s(-1) at 4 K.