The proceedings from the Abel Symposium on Geometry of Moduli, held at Svinøya Rorbuer, Svolvær in Lofoten, in August 2017, present both survey and research articles on the recent surge of developments in understanding moduli problems in algebraic geometry. Written by many of the main contributors to this evolving subject, the book provides a comprehensive collection of new methods and the various directions in which moduli theory is advancing. These include the geometry of moduli spaces, non-reductive geometric invariant theory, birational geometry, enumerative geometry, hyper-kähler geometry, syzygies of curves and Brill-Noether theory and stability conditions. Moduli theory is ubiquitous in algebraic geometry, and this is reflected in the list of moduli spaces addressed in this volume: sheaves on varieties, symmetric tensors, abelian differentials, (log) Calabi-Yau varieties, points on schemes, rational varieties, curves, abelian varieties and hyper-Kähler manifolds.

Arend Bayer: Bridgeland Stability on Kuznetsov Components in Families.- Jim Bryan: Donaldson-Thomas Invariants of the Banana Manifold and Elliptic Genera.- Ana-Maria Castravet: Derived Categories of Moduli Spaces of Stable Rational Curves.- Dawei Chen: Geometry of Moduli of Abelian Differentials.- Izzet Coskun: The Cohomology and Birational Geometry of Moduli Spaces of Sheaves on Surfaces.- Barbara Fantechi: Infinitesimal Deformations of log Calabi Yau Varieties and orb Ifolds.- Maksym Fedorchuk: Stability of Hilbert Points and Applications.- Brendan Hassett: Rationality in Families.- Klaus Hulek: Degenerations of Hilbert Schemes of Degree 0 Cycles on Surfaces.- Michael Kemeny: On the Possible Betti Tables of a Canonical Curve.- Frances Kirwan: Applications of Non-Reductive Geometric Invariant Theory.- Emanuele Macri: Bridgeland Stability and the Genus of Space Curves.- Kieran O'Grady: Abelian Varieties Associated to Hyperkählers of Kummer Type.-Andrei Okounkov: Monodromy and Derived Equivalences.- Aaron Pixton: Polynomiality of the Double Ramification Cycle.- Claire Voisin: Cubic Fourfolds, Hyper-Kähler Manifolds and their Degenerations.