A parallel advancing front scheme has been developed. The domain to be gridded is first subdivided spatially using a relatively coarse octree. Boxes are then identified and gridded in parallel. A scheme that resembles closely the advancing front technique on scalar machines is recovered by only considering the boxes of the active front that generate small elements. The procedure has been implemented on the SGI origin class of machines using the shared memory paradigm. Timings for a variety of cases show speedups similar to those obtained for flow codes. The procedure has been used to generate grids with tens of millions of elements.