There is a demand for the manufacture of two-dimensional (2D) materials with high-quality single crystals of large size. Usually, epitaxial growth is considered the method of choice1 in preparing single-crystalline thin films, but it requires single-crystal substrates for deposition. Here we present a different approach and report the synthesis of single-crystal-like monolayer graphene films on polycrystalline substrates. The technological realization of the proposed method resembles the Czochralski process and is based on the evolutionary selection2 approach, which is now realized in 2D geometry. The method relies on ‘self-selection’ of the fastest-growing domain orientation, which eventually overwhelms the slower-growing domains and yields a single-crystal continuous 2D film. Here we have used it to synthesize foot-long graphene films at rates up to 2.5 cm h−1 that possess the quality of a single crystal. We anticipate that the proposed approach could be readily adopted for the synthesis of other 2D materials and heterostructures.