This paper describes a volume rendering system for unstructured data, especially finite element data, that creates images with very high accuracy. The system will currently handle meshes whose cells are either linear or quadratic tetrahedra. Compromises or approximations are not introduced for the sake of efficiency. Whenever possible, exact mathematical solutions for the radiance integrals involved and for interpolation are used. The system will also handle meshes with mixed cell types: tetrahedra, bricks, prisms, wedges, and pyramids, but not with high accuracy. Accurate semi-transparent shaded isosurfaces may be embedded in the volume rendering. For very small cells, subpixel accumulation by splatting is used to avoid sampling error. A revision to an existing accurate visibility ordering algorithm is described, which includes a correction and a method for dramatically increasing its efficiency. Finally, hardware assisted projection and compositing are extended from tetrahedra to arbitrary convex polyhedra.