The question of how best to compare and classify the 3D structures of proteins is one of the most important unsolved problems in computational biology. To help tackle this problem, we have developed a novel shape-density superposition algorithm called 3D-Blast which represents and superposes the shapes of protein backbone folds using the spherical polar Fourier correlation technique originally developed by us for protein docking. The utility of this approach is compared with several well known protein structure alignment algorithms using receiver-operator-characteristic plots of queries against the "gold standard" CATH database. Despite being completely independent of protein sequences and using no information about the internal geometry of proteins, our results from searching the CATH database show that 3D-Blast is highly competitive compared to current state-of-the-art protein structure alignment algorithms. A novel and potentially very useful feature of our approach is that it allows an average or "consensus" fold to be calculated easily for a given group of protein structures. We find that using consensus shapes to represent entire fold families also gives very good database query performance. We propose that using the notion of consensus fold shapes could provide a powerful new way to index existing protein structure databases, and that it offers an objective way to cluster and classify all of the currently known folds in the protein universe.