Fast techniques for direct volume rendering over curvilinear grids of hexahedral cells are developed. This type of 3D grid is common in computational uid dynamics and nite element analysis. Four new projection methods are presented and compared with each other and with previous methods for tetrahedral grids and rectilinear grids. All four methods use polygon-rendering hardware for speed. A simpli ed algorithm for visibility ordering, which is based on a combination of breadth- rst and depth- rst searches, is described. A new multi-pass blending method is described that reduces visual artifacts that are introduced by linear interpolation in hardware where exponential interpolation is needed. Multipass blending is of equal interest to hardware-oriented projection methods used on rectilinear grids. Visualization tools that permit rapid data banding and cycling through transfer functions, as well as region restriction, are described. 1