The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and continues to be an important tool today. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits in 1962, places him as the founder of orbit theory. The origins of the orbit method lie in the search for a relationship between classical and quantum mechanics. Over the years, the orbit method has been used to link harmonic analysis (theory of unitary representations of Lie groups) with differential geometry (symplectic geometry of homogeneous spaces), and it has stimulated and invigorated many classical domains of mathematics, i.e., representation theory, integrable systems, complex algebraic geometry, to name several. It continues to be a useful and powerful tool in all of these areas of mathematics and mathematical physics.This volume, dedicated to A. A. Kirillov, covers a very broad range of key topics such as:* The orbit method in the theory of unitary representations of Lie groups* Infinite-dimensional Lie groups: their orbits and representations* Quantization and the orbit method; geometric quantization (old and new)* The Virasoro algebra; string and conformal field theories* Lie superalgebras and their representations* Combinatorial aspects of representation theory.The prominent contributors to this volume present original and expository invited papers in the areas of Lie theory, geometry, algebra, and mathematical physics. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses.Contributors include: A. Alekseev, J. Alev, R. Brylinski, J. Dixmier, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina.