Flexible materials and devices could be exploited in light-emitting diodes, electronic circuits, memory devices, sensors, displays, solar cells and bioelectronic devices. Nanoscale elements such as thin films, nanowires, nanotubes and nanoparticles can also be incorporated into the active films of mechanically flexible devices. Large-area devices containing extremely thin films of molecular materials represent the ultimate scaling of flexible devices based on organic materials, but the influence of bending and twisting on the electrical and mechanical stability of such devices has never been examined. Here, we report the fabrication and characterization of two-terminal electronic devices based on self-assembled monolayers of alkyl or aromatic thiol molecules on flexible substrates. We find that the charge transport characteristics of the devices remain stable under severe bending conditions (radius ≤ 1 mm) and a large number of repetitive bending cycles (≥1,000). The devices also remain reliable in various bending configurations, including twisted and helical structures.