Compression techniques for triangle-mesh representations of 3D models have been the focus of many recent efforts from the graphics, modeling, and theory research community; from developers of computer graphics hardware and software; and from organizations that define international standards. An ideal compression technology would simultaneously support the following three objectives: (1) progressive refinements of the received mesh during decompression, (2) nearly optimal compression ratios for both geometry and connectivity, and (3) in-line, real-time decompression algorithms for hardware or software implementations. Because these three objectives impose contradictory constraints on the compressed format, previously reported efforts focus primarily on one - sometimes two - of these objectives. The SQUEEZE technique introduced here for Fast and Progressive Decompression of Triangle Meshes addresses all three constraints simultaneously and attempts to provide the best possible compromise for the needs of common internet applications that require frequent access to remote 3D data-bases. For a typical mesh of T triangles, SQUEEZE compresses the connectivity to 3.7T bits, which is competitive with the best progressive compression techniques reported so far. The geometry prediction encoding techniques introduced here lead to an additional 20% improvement in geometry compression over previous schemes. Our initial implementation on a 200 Mhz CPU achieves a decompression rate of about 15000 triangles per second. Finally, in general SQUEEZE downloads a model through 10 successive refinement stages, providing the full benefit of progressivity. After each refinement step, the user may manipulate the current resolution model as SQUEEZE decompresses the next upgrade, or temporarily stop the transmission until a higher level-of-detail is needed.