W-symmetry is an extension of conformal symmetry in two dimensions. Since its introduction in 1985, W-symmetry has become one of the central notions in the study of two-dimensional conformal field theory. The mathematical structures that underlie W-symmetry are so-called W-algebras, which are higher-spin extensions of the Virasoro algebra. This book contains a collection of papers on W-symmetry, covering the period from 1985 through 1992. Its main focus is the construction of W-algebras and their representation theory. A recurrent theme is the intimate connection between W-algebras and affine Lie algebras. Some of the applications, in particular W-gravity, are also covered The Underlying Geometry of the Fixed Centers Problems, A. Albouy; Critical Equations for the Polyharmonic Operator, T. Bartsch; Heat Method in Nonlinear Elliptic Equations, K-C. Chang; Boundary Blow-Up Solutions and their Applications, Y.H. Du; Fixed Points of Increasing Operator, F.Y. Li; Collinear Central Configurations in Celestial Mechanics, Y.M. Long, S.Z. Sun; Remarks on a Priori Estimates for Superlinear Elliptic Problems, M. Ramos; Multi-Parameter Schr dinger Equation with Magnetic Field, A. Szulkin; Sign Changing Solutions of Superlinear Schrodinger Equations, T. Weth; Computational Theory and Methods for Finding Multiple Critical Points, J.X. Zhou; and other papers