Two-dimensional (2D) materials have been studied extensively as monolayers(1-5), vertical or lateral heterostructures(6-8). To achieve functionalization, monolayers are often patterned using soft lithography and selectively decorated with molecules(9,10). Here we demonstrate the growth of a family of 2D materials that are intrinsically patterned. We demonstrate that a monolayer of PtSe2 can be grown on a Pt substrate in the form of a triangular pattern of alternating 1T and 1H phases. Moreover, we show that, in a monolayer of CuSe grown on a Cu substrate, strain relaxation leads to periodic patterns of triangular nanopores with uniform size. Adsorption of different species at preferred pattern sites is also achieved, demonstrating that these materials can serve as templates for selective self-assembly of molecules or nanoclusters, as well as for the functionalization of the same substrate with two different species.