This chapter provides a summary of the major properties of Siegel moduli schemes which will be used in later chapters. We have included a large amount of definitions and constructions to make this sketchy review reasonably self-contained. Thus we begin by describing geometric invariant theoretic construction of the Siegel moduli schemes, which also endows natural integral structure to these friendly moduli schemes. Then we turn to look at the picture over C, and construct the Satake compactification, which comprises the content of §§ 2 and 3. In §4 we introduce the theta functions, which provides canonical coordinates for our moduli schemes. Toroidal blowing up of the Satake compactifications and Tai's theorem on projectivity of these nice blow-ups are explained in §§5 and 6, which will satisfy anybody living in charateristic 0. Of course, the main goal of this thesis is to construct equisingular toroidal compactifications (over Z[½]).

Due to the immensity of the subject, no proof is provided for the nontrivial assertions made in this chapter. Instead, the readers are invited to take them on faith, or turn to the relevant literature cited at the end of this book.