I. Forming geometrical intuition statement of the main problem.- 1. Formulating the problem.- 2. Spherical geometry.- 3. Geometry on a cylinder.- 3.1. First acquaintance.- 3.2. How to measure distances.- 3.3. The study of geometry on a cylinder.- 4. A world in which right and left are indistinguishable.- 5. A bounded world.- 5.1. Description of the geometry.- 5.2. Lines on the torus.- 5.3. Some applications.- 6. What does it mean to specify a geometry?.- 6.1. The definition of a geometry.- 6.2. Superposing geometries.- II. The theory of 2-dimensional locally Euclidean geometries.- 7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- 7.1. Definition of equivalence by means of motions.- 7.2. The geometry corresponding to a uniformly discontinuous group.- 8. Classification of all uniformly discontinuous groups of motions of the plane.- 8.1. Motions of the plane.- 8.2. Classification: generalities and groups of Type I and II.- 8.3. Classification: groups of Type III.- 9. A new geometry.- 10. Classification of all 2-dimensional locally Euclidean geometries.- 10.1. Constructions in an arbitrary geometry.- 10.2. Coverings.- 10.3. Construction of the covering.- 10.4. Construction of the group.- 10.5. Conclusion of the proof of Theorem 1.- III. Generalisations and applications.- 11. 3-dimensional locally Euclidean geometries.- 11.1. Motions of 3-space.- 11.2. Uniformly discontinuous groups in 3-space: generalities.- 11.3. Uniformly discontinuous groups in 3-space: classification.- 11.4. Orientability of the geometries.- 12. Crystallographic groups and discrete groups.- 12.1. Symmetry groups.- 12.2. Crystals and crystallographic groups.- 12.3. Crystallographic groups and geometries: discrete groups.- 12.4. A typical example: the geometry of the rectangle.- 12.5. Classification of all locally Cn or Dn geometries.- 12.6. On the proof of Theorems 1 and 2.- 12.7. Crystals and their molecules.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- 13. Similarity of geometries.- 13.1. When are two geometries defined by uniformly discontinuous groups the same?.- 13.2. Similarity of geometries.- 14. Geometries on the torus.- 14.1. Geometries on the torus and the modular figure.- 14.2. When do two pairs of vectors generate the same lattice?.- 14.3. Application to number theory.- 15. The algebra of similarities: complex numbers.- 15.1. The geometrical definition of complex numbers.- 15.2. Similarity of lattices and the modular group.- 16. Lobachevsky geometry.- 16.1. 'Motions'.- 16.2. 'Lines'.- 16.3. Distance.- 16.4. Construction of the geometry concluded.- 17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- 17.1. Discreteness of the modular group.- 17.2. The set of all geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.