1.Euclidean Geometry — "Old and New Euclidean Geometry and Analysis" 2. Transition — "The Need for a More General Framework" 3. Surfaces from Gauss to Today 4. Riemann's Blueprints — "Riemann's Blueprints for Architecture in Myriad Dimensions" 5. A One Page Panorama 6. Metric Geometry and Curvature — "Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature" 7. Volume and Inequalities on Volumes of Cycles 8. Transition: The Next Two Chapters 9.Spectrum of the Laplacian — "Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian" 10. Geodesic Dynamics — "Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics" 11. Best Metric — "What is the Best Riemannian Metric on a Compact Manifold?" 12. From Curvature to Topology 13. Holonomy Groups and Kähler Manifolds 14. Some Other Important Topics 15. The Technical Chapter