Toroidal groups are the connecting link between torus groups and any complex Lie groups. Many properties of complex Lie groups such as the pseudoconvexity and cohomology are determined by their maximal toroidal subgroups. Quasi-Abelian varieties are meromorphically separable toroidal groups. They are the natural generalisation of the Abelian varieties. Nevertheless, their behavior can be completely different as the wild groups show. TOC:The concept of toroidal groups: Irrationality and toroidal coordinates; Toroidal subgroups and pseudoconvexity. Line bundles and cohomology: Line bundles on toroidal groups; Comohology of toroidal groups. Quasi-Abelian Varieties: Ample Riemann forms; Characterisation of quasi-Abelian varieties. Reduction and extension: Automorphic forms; Extendable line bundles.

The Concept of Toroidal Groups.- Line Bundles and Cohomology.- Quasi-Abelian Varieties.- Reduction and Extension.