Using a geometric perspective, Protein Geometry, Classification, Topology, and Symmetry reviews and analyzes the structural principals of proteins with the goal of revealing the underlying regularities in their construction. It also reviews computer methods for structure analysis and the automatic comparison and classification of these structures with an analysis of the statistical significance of comparing different shapes. Following an analysis of the current state of protein classification, the authors explore more abstract geometric and topological representations, including the occurrence of knotted topologies. The book concludes with a consideration of the origin of higher-level symmetries in protein structure. The authors focus on simple geometric methods that are deterministic rather than probabilistic and on the more abstract simplifications of protein structure that allow a better understanding of the overall fold of the structure. Most of the methods described in this book have corresponding computer programs. These can be found (as C source code) at the ftp site of the Division of Mathematical Biology at the National Institute for Medical Research. This collection of ideas contains pedagogical material that make it ideal for post-graduate courses as well as new ideas and results essential for researchers investigating protein structures.