Toroidal Compactification of Siegel Spaces The siegel upperhalf plane and the symplectic group Main problem and main results The case of g=1 Boundary components and the structure of parabolic subgroups Realization as a siegel domain of the third kind and satake compactification Theory of torus embeddings Toroidal compactification due to Mumford Examples : reduction theory of positive quadratic forms An application of the Voronoi compactification to the theory of moduli of polarized abelian varieties.