This volume is a collection of surveys on function theory ineuclidean n-dimensional spaces centered around the theme ofquasiconformal space mappings. These surveys cover or arerelated to several topics including inequalities forconformal invariants and extremal length, distortiontheorems, L(p)-theory of quasiconformal maps, nonlinearpotential theory, variational calculus, value distributiontheory of quasiregular maps, topological properties ofdiscrete open mappings, the action of quasiconformal maps inspecial classes of domains, and global injectivity theorems.The present volume is the first collection of surveys onQuasiconformal Space Mappings since the origin of the theoryin 1960 and this collection provides in compact form accessto a wide spectrum of recent results due to well-knownspecialists.CONTENTS: G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen:Conformal invariants, quasiconformal maps and specialfunctions.- F.W. Gehring: Topics in quasiconformalmappings.- T.Iwaniec: L(p)-theory of quasiregularmappings.- O. Martio: Partial differential equations andquasiregular mappings.- Yu.G. Reshetnyak: On functionalclasses invariant relative to homothetics.- S. Rickman:Picard's theorem and defect relation for quasiconformalmappings.- U. Srebro: Topological properties of quasiregularmappings.- J. V{{is{{l{{: Domains and maps.- V.A. Zorich: Theglobal homeomorphism theorem for space quasiconformalmappings, its development and related open problems.