We present a new approach to designing data structures for the important problem of externalmemory range searching in two and three dimensions. We construct data structures for answering range queries in O((log log log B N) log B N + K=B) I/O operations, where N is the number of points in the data structure, B is the I/O block size, and K is the number of points in the answer to the query. Our data structures answer a longstanding open problem by providing three dimensional results comparable to those provided by [8, 10] for the two dimensional case, though completely new techniques are used. Ours is the first 3-D range search data structure that simultaneously achieves both a base-B logarithmic search overhead (namely, (log log log B N) log B N) and a fully blocked output component (namely, K=B). This gives us an overall I/O complexity extremely close to the well-known lower bound of \Omega\Gamma/89 B N +K=B). We base our data structures on the novel concept of B-approximate boundarie..
