Carolina Araujo is a researcher at the Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil. Ana-Maria Castravet is Professor at the University of Versailles, France. Ivan Cheltsov is Chair of Birational Geometry at the University of Edinburgh. Kento Fujita is Associate Professor at Osaka University. Anne-Sophie Kaloghiros is a Reader at Brunel University London. Jesus Martinez-Garcia is Senior Lecturer in Pure Mathematics at the University of Essex. Constantin Shramov is a researcher at the Steklov Mathematical Institute, Moscow. Hendrik Süß is Chair of Algebra at the University of Jena, Germany. Nivedita Viswanathan is a Research Associate at Loughborough University.

'The notion of K-stability for Fano manifold has origins in differential geometry and geometric analysis but is now also of fundamental importance in algebraic geometry, with recent developments in moduli theory. This monograph gives an account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105 deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London

• Introduction; 1. K-stability; 2. Warm-up: smooth del Pezzo surfaces; 3. Proof of main theorem: known cases; 4. Proof of main theorem: special cases; 5. Proof of main theorem: remaining cases; 6. The big table; 7. Conclusion; Appendix. Technical results used in proof of main theorem; References; Index.

Carolina Araujo is a researcher at the Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil. Ana-Maria Castravet is Professor at the University of Versailles, France. Ivan Cheltsov is Chair of Birational Geometry at the University of Edinburgh. Kento Fujita is Associate Professor at Osaka University. Anne-Sophie Kaloghiros is a Reader at Brunel University London. Jesus Martinez-Garcia is Senior Lecturer in Pure Mathematics at the University of Essex. Constantin Shramov is a researcher at the Steklov Mathematical Institute, Moscow. Hendrik Süß is Chair of Algebra at the University of Jena, Germany. Nivedita Viswanathan is a Research Associate at Loughborough University.

'The notion of K-stability for Fano manifold has origins in differential geometry and geometric analysis but is now also of fundamental importance in algebraic geometry, with recent developments in moduli theory. This monograph gives an account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105 deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London

• Introduction; 1. K-stability; 2. Warm-up: smooth del Pezzo surfaces; 3. Proof of main theorem: known cases; 4. Proof of main theorem: special cases; 5. Proof of main theorem: remaining cases; 6. The big table; 7. Conclusion; Appendix. Technical results used in proof of main theorem; References; Index.

Carolina Araujo is a researcher at the Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil. Ana-Maria Castravet is Professor at the University of Versailles, France. Ivan Cheltsov is Chair of Birational Geometry at the University of Edinburgh. Kento Fujita is Associate Professor at Osaka University. Anne-Sophie Kaloghiros is a Reader at Brunel University London. Jesus Martinez-Garcia is Senior Lecturer in Pure Mathematics at the University of Essex. Constantin Shramov is a researcher at the Steklov Mathematical Institute, Moscow. Hendrik Süß is Chair of Algebra at the University of Jena, Germany. Nivedita Viswanathan is a Research Associate at Loughborough University.

'The notion of K-stability for Fano manifold has origins in differential geometry and geometric analysis but is now also of fundamental importance in algebraic geometry, with recent developments in moduli theory. This monograph gives an account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105 deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London

• Introduction; 1. K-stability; 2. Warm-up: smooth del Pezzo surfaces; 3. Proof of main theorem: known cases; 4. Proof of main theorem: special cases; 5. Proof of main theorem: remaining cases; 6. The big table; 7. Conclusion; Appendix. Technical results used in proof of main theorem; References; Index.