The paper reports on a scanning tunneling microscopy ͑STM͒ study and computer simulation with an N-body interatomic interaction potential of a graphite monolayer ͑i.e., graphene͒ on the Ni͑111͒, ͑110͒, ͑755͒, and ͑771͒ single-crystal surfaces. Unlike the case of graphene on Ni͑111͒, which forms a solid single-crystal coating with ͑1 ϫ 1͒ structure, graphene on the Ni͑110͒ surface forms a complex crystal structure distorted substantially by interaction with the substrate. Calculations of the graphene/Ni͑110͒ system have revealed that the strong chemical interaction of carbon with nickel gives rise to a noticeable curving of the graphene layer on a scale of a few angstroms. The model thus derived has permitted proper interpretation of the experimental data obtained by STM, as well as prediction of the main result of studies of graphene formed on faceted surfaces, which have revealed the ability of graphene to coat geometrically nonuniform surfaces in the form of a curved continuous film.