This paper presents a novel sequence-independent method of aligning protein structures using three-dimensional spherical polar Fourier (SPF) representations of protein shape. The approach is demonstrated by clustering subsets of the CATH database for each of the four main CATH fold types, and by searching the entire CATH database of some 12,000 structures using several protein structures as queries. Overall, the automatic SPF clustering approach agrees very well with the expert-curated CATH classification, and ROC plot analyses of the database searches show that the approach has very high precision and recall. Database query times can be reduced considerably by using a simple rotationally-invariant pre-filter in tandem with a more sensitive rotational search with little or no reduction in accuracy. Hence it should soon be possible to perform on-line 3D structural searches in interactive time-scales.