It is the purpose of this paper to survey much of what is known about a special type of contact structure,namely, a Sasakian structure. I concentrate on results obtained since the publication of the book . More-over, results in the book are placed in the context of Sasaki moduli spaces.Roughly, Sasakian geometry is to contact geometry what Kählerian geometry is to symplectic geometry.
A contact structureDis said to be ofSasaki typeif there is a Sasakian structureSwhose contact 1-formηsatisesD= kerη. Equivalently, the foliationFRdescribed by the Reeb vector eldRis Kählerian andRliesinaut(S), the Lie algebra of the group of Sasaki automorphisms. Moreover, the ane cone(C(M),I)associatedto a Sasaki manifoldMhas a natural exact Kählerian structure. Thus, Sasaki geometry, sandwiched naturallybetween two Kähler geometries, is considered to be the odd dimensional sister to Kähler geometry.