Previous studies of the interaction of water with graphene-coated surfaces have been limited to flat (smooth) surfaces. Here we created a rough surface by nanopatterning and then draped the surface with a single-layer graphene sheet. We found that the ultrasheer graphene drape prevents the penetration of water into the textured surface thereby drastically reducing the contact angle hysteresis (which is a measure of frictional energy dissipation) and preventing the liquid contact line from getting pinned to the substrate. This has important technological implications since the main obstacle to the motion of liquid drops on rough surfaces is contact angle hysteresis and contact line pinning. Graphene drapes could therefore enable enhanced droplet mobility which is required in a wide range of applications in micro and nanofluidics. Compared to polymer coatings that could fill the cavities between the nano/micropores or significantly alter the roughness profile of the substrate, graphene provides the thinnest (i.e., most sheer) and most conformal drape that is imaginable. Despite its extreme thinness, the graphene drape is mechanically robust, chemically stable, and offers high flexibility and resilience which can enable it to reliably drape arbitrarily complex surface topologies. Graphene drapes may therefore provide a hitherto unavailable ability to tailor the dynamic wettability of surfaces for a variety of applications.