Ultra perfect graphene monolayers, islands and ribbons can be epitaxially grown on different single crystal metal surfaces under Ultra High Vacuum conditions. These graphene layers are spontaneously nanostructured in a periodic array of ripples by the Moiré patterns caused by the difference in lattice parameter with the different substrates. There are, however, important differences among them. In-situ Scanning Tunneling Microscopy of graphene monolayers on Ru (0001) or Ir (111), for instance, shows that the apparent corrugation is much larger for gr/Ru than for gr/Ir. For gr/Ru, it depends strongly on the bias voltage and can even be inverted above+ 2.6 V, revealing that, in addition to the geometric corrugation, a much stronger electronic corrugation exists. The combination of He diffraction and high resolution STM allows us to determine the epitaxial relationship between graphene and Ru lattices, quantify the …