Abelian varieties can be classified via their moduli. In positive
characteristic the structure of the p-torsion-structure is an
additional, useful tool. For that structure supersingular abelian
varieties can be considered the most special ones. They provide a
starting point for the fine description of various structures. For
low dimensions the moduli of supersingular abelian varieties is by
now well understood. In this book we provide a description of the
supersingular locus in all dimensions, in particular we compute the
dimension of it: it turns out to be equal to g.g/4, and we express
the number of components as a class number, thus completing a long
historical line where special cases were studied and general
results were conjectured (Deuring, Hasse, Igusa, Oda-Oort,
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