Developments and Trends in Infinite Dimensional Lie Theory This collection of invited expository articles focuses on recent developments and trends in infinite dimensional Lie theory which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite dimensional Lie (super )algebras geometry of infinite dimensional Lie (transformation) groups and representation theory of infinite dimensional Lie groups. Part (A) is mainly concerned with the structure and representation theory of infinite dimensional Lie algebras and contains articles on the structure of direct limit Lie algebras extended affine Lie algebras and loop algebras as well as representations of loop algebras and Kac–Moody superalgebras. The articles in Part (B) examine connections between infinite dimensional Lie theory and geometry. The topics range from infinite dimensional groups acting on fiber bundles corresponding characteristic classes and gerbes to Jordan theoretic geometries and new results on direct limit groups. The analytic representation theory of infinite dimensional Lie groups is still very much underdeveloped. The articles in Part (C) develop new promising methods based on heat kernels multiplicity freeness Banach–Lie–Poisson spaces and infinite dimensional generalizations of reductive Lie groups. Contributors: B. Allison D. Beltiţă W. Bertram J. Faulkner Ph. Gille H. Glöckner K. H. Neeb E. Neher I. Penkov A. Pianzola D. Pickrell T.S. Ratiu N.R. Scheithauer C. Schweigert V. Serganova K. Styrkas K. Waldorf and J.A. Wolf. Preface Part A: Infinite Dimensional Lie (Super )Algebras Isotopy for Extended Affine Lie Algebras and Lie Tori Remarks on the Isotriviality of Multiloop Algebras Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey Tensor Representations of Classical Locally Finite Lie Algebras Lie Algebras Vertex Algebras and Automorphic Forms Kac–Moody Superalgebras and Integrability Part B: Geometry of Infinite Dimensional Lie (Transformation) Groups Jordan Structures and Non Associative Geometry Direct Limits of Infinite Dimensional Lie Groups Lie Groups of Bundle Automorphisms and Their Extensions Gerbes and Lie Groups Part C: Representation Theory of Infinite Dimensional Lie Groups Functional Analytic Background for a Theory of Infinite Dimensional Reductive Lie Groups Heat Kernel Measures and Critical Limits Coadjoint Orbits and the Beginnings of a Geometric Representation Theory Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces Index.