Accurate comparisons of different protein structures play important roles in structural biology, structure prediction and functional annotation. The root-mean-square-deviation (RMSD) after optimal superposition is the predominant measure of similarity due to the ease and speed of computation. However, global RMSD is dependent on the length of the protein and can be dominated by divergent loops that can obscure local regions of similarity. A more sophisticated measure of structure similarity, Template Modeling (TM)-score, avoids these problems, and it is one of the measures used by the community-wide experiments of critical assessment of protein structure prediction to compare predicted models with experimental structures. TM-score calculations are, however, much slower than RMSD calculations. We have therefore implemented a very fast version of TM-score for Graphical Processing Units (TM-score-GPU), using a new and novel hybrid Kabsch/quaternion method for calculating the optimal superposition and RMSD that is designed for parallel applications. This acceleration in speed allows TM-score to be used efficiently in computationally intensive applications such as for clustering of protein models and genome-wide comparisons of structure.