History -- The Revolution of János Bolyai -- Gauss and Non-Euclidean Geometry -- János Bolyai’s New Face -- Axiomatical and Logical Aspects -- Hyperbolic Geometry, Dimension-Free -- An Absolute Property of Four Mutually Tangent Circles -- Remembering Donald Coxeter -- Axiomatizations of Hyperbolic and Absolute Geometries -- Logical Axiomatizations of Space-Time. Samples from the Literature -- Polyhedra, Volumes, Discrete Arrangements, Fractals -- Structures in Hyperbolic Space -- The Symmetry of Optimally Dense Packings -- Flexible Octahedra in the Hyperbolic Space -- Fractal Geometry on Hyperbolic Manifolds -- A Volume Formula for Generalised Hyperbolic Tetrahedra -- Tilings, Orbifolds and Manifolds, Visualization -- The Geometry of Hyperbolic Manifolds of Dimension at Least 4 -- Real-Time Animation in Hyperbolic, Spherical, and Product Geometries -- On Spontaneous Surgery on Knots and Links -- Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces -- Differential Geometry -- Non-Euclidean Analysis -- Holonomy, Geometry and Topology of Manifolds with Grassmann Structure -- Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry -- How Far Does Hyperbolic Geometry Generalize? -- Geometry of the Point Finsler Spaces -- Physics -- Black Hole Perturbations -- Placing the Hyperbolic Geometry of Bolyai and Lobachevsky Centrally in Special Relativity Theory: An Idea Whose Time has Returned.