Tritium containment and control are essential prerequisites for fusion to comply with the promise of a safe energy with low impact on the environment. In particular, the behaviour of tritium in materials, i.e its adsorption, absorption, diffusion and trapping parameters, has to be assessed. The multi-layer design of DEMO plasma-facing components as well as the foreseen breeding blankets design includes several materials and interfaces, which behaviour regarding permeation remains to be clarified. The surface processes on both sides (plasma and cooling loop) also require more investigation. In this work, hydrogen transport parameters (diffusion and trapping) are determined using gas-driven permeation experiments undertaken on Eurofer 97, 316L stainless steel, copper and CuCrZr. The purpose of the presentation is to introduce the Hydrogen PERmeation in TOkamak-relevant MATErials (Hypertomate) experiment and the subsequent results on hydrogen permeation and trapping. Eurofer97 results will be recalled first. We will show that they are in good agreement with literature in the case of a perfect material. Particular attention has been given to the effect of surface state (roughness and oxidation) that influences the permeation behaviour. The impact of roughness on permeation results will be exemplified. This permeation analysis is supplemented with thermal desorption spectroscopy performed on deuterium-loaded samples in order to investigate more thoroughly trapping in these materials. These results altogether feed our 3D finite element code which aim is to fully simulate permeation in the plasma-facing components or breeding blankets. The ongoing developments to carry out the same measurements using tritium as a driving gas will also be presented. These measurements will show whether or not tritium differs from hydrogen or deuterium, for instance because of surface effects induced by its beta emission.