Structure comparison opens a window into the distant past of protein evolution, which has been unreachable by sequence comparison alone. With 55,000 entries in the Protein Data Bank and about 500 new structures added each week, automated processing, comparison, and classification are necessary. A variety of methods use different representations, scoring functions, and optimization algorithms, and they generate contradictory results even for moderately distant structures. Sequence mutations, insertions, and deletions are accommodated by plastic deformations of the common core, retaining the precise geometry of the active site, and peripheral regions may refold completely. Therefore structure comparison methods that allow for flexibility and plasticity generate the most biologically meaningful alignments. Active research directions include both the search for fold invariant features and the modeling of structural transitions in evolution. Advances have been made in algorithmic robustness, multiple alignment, and speeding up database searches.