Protein structures are flexible and undergo structural rearrangements as part of their function, and yet most existing protein structure comparison methods treat them as rigid bodies, which may lead to incorrect alignment. We have developed the Flexible structure AlignmenT by Chaining AFPs (Aligned Fragment Pairs) with Twists (FATCAT), a new method for structural alignment of proteins. The FATCAT approach simultaneously addresses the two major goals of flexible structure alignment; optimizing the alignment and minimizing the number of rigid-body movements (twists) around pivot points (hinges) introduced in the reference protein. In contrast, currently existing flexible structure alignment programs treat the hinge detection as a post-process of a standard rigid body alignment. We illustrate the advantages of the FATCAT approach by several examples of comparison between proteins known to adopt different conformations, where the FATCAT algorithm achieves more accurate structure alignments than current methods, while at the same time introducing fewer hinges.