Magnetic skyrmions are chiral spin structures with a whirling configuration. Their topological properties, nanometre size and the fact that they can be moved by small current densities have opened a new paradigm for the manipulation of magnetization at the nanoscale. Chiral skyrmion structures have so far been experimentally demonstrated only in bulk materials and in epitaxial ultrathin films, and under an external magnetic field or at low temperature. Here, we report on the observation of stable skyrmions in sputtered ultrathin Pt/Co/MgO nanostructures at room temperature and zero external magnetic field. We use high lateral resolution X-ray magnetic circular dichroism microscopy to image their chiral Néel internal structure, which we explain as due to the large strength of the Dzyaloshinskii-Moriya interaction as revealed by spin wave spectroscopy measurements. Our results are substantiated by micromagnetic simulations and numerical models, which allow the identification of the physical mechanisms governing the size and stability of the skyrmions.