Lie linksbetweentorus and Toroidal arethe complex missing groups any groups such and of Lie as complex pseudoconvexity groups. Manyphenomena groups the of beunderstood thestructure can onlythrough concept cohomologygroups of different behavior ofthe oftoroidal The cohomology complex groups groups. the of their toroidal Lie be characterized can by properties groups - groups in their centers. pearing book. So the oldest have not been treated in a Toroidal systematically groups in it who worked in this field and the mathematician youngest working living aboutthemain results these decidedto a concerning comprehensivesurvey give and to discuss problems. open groups of the torus As the Toroidal are generalization groups. groups non-compact and Grauert. As in the sense ofAndreotti manifolds are convex complex they others have similarbehaviorto Lie someofthem a complextori, complex groups whencec- different with for non-Hausdorff are example cohomology groups, mustbe used. newmethods pletely of is to describe the fundamental The aim of these lecture notes properties the reductiontheorem toroidal As a result ofthe qua- meromorphic groups. basic ends inthethird varieties of interest.Their Abelian are special description MainTheorem. withthe chapter wide atthe - ofSOPHus LIE -wasintroducedtoa This inhonour public theory " 1999. after Lie" in on Conference 100Years Leipzig, July 8-9, Sophus HUMBOLDT wishes to thank the ALEXANDER VON The first-named author FOUNDATION for partial support. December 1998 Hannoverand Toyama, YukitakaAbeandKlaus Kopfermann Contents 1 Introduction ..................................................... of Toroidal 3 1. The Concept Groups ............................. 1.1 and toroidal coordinates 3 Irrationality ........................ Toroidal 3 ........................................... groups 7 Complex homomorphisms .................................. Toroidal coordinates and C*n-q -fibre bundles 9 .................