In this topical review we describe the fabrication, characterization and applications of 1 nm thick, mechanically stable carbon nanomembranes (CNMs). They represent a new type of functional two-dimensional (2D) materials, which can be concisely described as “surfaces without bulk”. Because CNMs are made by electron-induced crosslinking of aromatic self-assembled monolayers (SAMs), we start with an overview of SAMs with a special emphasis on aromatic SAMs. We describe the chemical modification of SAMs by electron, ion and photon irradiation, introduce the concepts of irradiation-induced crosslinking and chemical nanolithography of aromatic SAMs and discuss the underlying physical and chemical mechanisms. We present examples for applications of these phenomena in the engineering of complex surface architectures, e.g., nanopatterns of proteins, fluorescent dyes or polymer brushes. Then we introduce a transfer procedure to release cross-linked aromatic SAMs from their original substrates and to form free-standing CNMs. We discuss mechanical and electrical properties of CNMs and demonstrate that they can be converted into graphene upon annealing. This transformation opens an original and flexible molecular route towards the large-scale synthesis of graphene sheets with tunable properties. Finally, we demonstrate the lithographic and chemical tailoring of CNMs to fabricate novel functional 2D carbon materials: supports for high resolution transmission electron microscopy (HRTEM) and nanolithography, nanosieves, Janus nanomembranes, polymer carpets, complex layered structures. Prospects of combining different types of nanomembranes made of SAMs (CNMs, graphene, nanosieves, Janus nanomembranes) towards the engineering of novel functional nanomaterials for a variety of electronic, optical, lab-on-a-chip and micro-/nanomechanical (MEMS/NEMS) devices are discussed.