In this paper we present a tetrahedron-based, h-re®nement-type algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell-centred, Riemann problem-based, ®nite volume scheme of the MUSCL type, employing an approximate Riemann solver. The adaptive scheme is then used to compute the diffraction of shock waves around a box section corner for subsonic and supersonic post-shock ¯ow. In the subsonic case, preliminary measurements of vortex ®lament speed and vortical Mach number are in broad quantitative agreement with known theoretical results.