We present a new architecture for interactive unstructured volume rendering. Our system moves all the computations necessary for order-independent transparency and volume scan conversion from the CPU to the graphics hardware, and it makes a software sorting pass unnecessary. It therefore provides the same advantages for volume data that triangle- processing hardware provides for surfaces. To address a remaining bottleneck - the bandwidth between main memory and the graphics processor - we introduce two new primitives, tetrahedral strips and tetrahedral fans. These primitives allow performance improvements in rendering tetrahedral meshes similar to the improvements triangle strips and fans allow in rendering triangle meshes. We provide new techniques for generating tetrahedral strips that achieve, on the average, strip lengths of 17 on representative datasets. The combined effect of our architecture and new primitives is a 72 to 85 times increase in performance over triangle graphics hardware approaches. These improvements make it possible to use volumetric tetrahedral meshes in interactive applications. Notes: Presented at and published in the International Workshop on Volume Graphics, June 21-22, 2001, Stony Brook, New York 17 Pages