From a geometric perspective, this book 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 the 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.