Free-standing nanomembranes with molecular or atomic thickness are currently explored for separation technologies, electronics, and sensing. Their engineering with well-defined structural and functional properties is a challenge for materials research. Here we present a broadly applicable scheme to create mechanically stable carbon nanomembranes (CNMs) with a thickness of ~0.5 to ~3 nm. Monolayers of polyaromatic molecules (oligophenyls, hexaphenylbenzene, and polycyclic aromatic hydrocarbons) were assembled and exposed to electrons that cross-link them into CNMs; subsequent pyrolysis converts the CNMs into graphene sheets. In this transformation the thickness, porosity, and surface functionality of the nanomembranes are determined by the monolayers, and structural and functional features are passed on from the molecules through their monolayers to the CNMs and finally on to the graphene. Our procedure is scalable to large areas and allows the engineering of ultrathin nanomembranes by controlling the composition and structure of precursor molecules and their monolayers.