The structure of a shock wave in a vibrationally relaxing gas undergoing reflexion from a plane wall is examined. The shock wave is assumed to be weak, and departures from thermodynamic equilibrium are assumed small; both an adiabatic and an isothermal wall are considered. The flow field is divided into three regions: a far-field region, an interaction region, and, for the isothermal-wall case, a thermal boundary layer. Different asymptotic expansions are determined for the various regions through the method of matched asymptotic expansions. In the region far from the wall, a non-equilibrium Burgers equation governs the motion and the incident and the reflected shock wave structures. During reflexion, a non-equilibrium wave equation applies; its first-order terms are equivalent to an acoustic approximation. Heat conduction to the wall is modelled by an isothermal wall boundary condition which requires the introduction of a thermal boundary layer adjacent to the wall. This thermal boundary layer is thin and the adiabatic-wall result provides the outer solution for treating this layer. This thermal layer affects the structure of the reflected wave.