Intro; Preface; Early history of Alexandov geometry; Manifesto of Alexandrov geometry; Acknowledgements; Contents; 1 Preliminaries; 1.1 Metric spaces; 1.2 Constructions; 1.3 Geodesics, triangles, and hinges; 1.4 Length spaces; 1.5 Model angles and triangles; 1.6 Angles and the first variation; 1.7 Space of directions and tangent space; 1.8 Hausdorff convergence; 1.9 Gromov-Hausdorff convergence; 2 Gluing theorem and billiards; 2.1 The 4-point condition; 2.2 Thin triangles; 2.3 Reshetnyak's gluing theorem; 2.4 Reshetnyak puff pastry; 2.5 Wide corners; 2.6 Billiards; 2.7 Comments

3 Globalization and asphericity3.1 Locally CAT spaces; 3.2 Space of local geodesic paths; 3.3 Globalization; 3.4 Polyhedral spaces; 3.5 Flag complexes; 3.6 Cubical complexes; 3.7 Exotic aspherical manifolds; 3.8 Comments; 4 Subsets; 4.1 Motivating examples; 4.2 Two-convexity; 4.3 Sets with smooth boundary; 4.4 Open plane sets; 4.5 Shefel's theorem; 4.6 Polyhedral case; 4.7 Two-convex hulls; 4.8 Proof of Shefel's theorem; 4.9 Comments; Semisolutions; References; ; Index