This is a survey of the relationship between C*-algebraic deformation
quantization and the tangent groupoid in noncommutative geometry, emphasizing
the role of index theory. We first explain how C*-algebraic versions of
deformation quantization are related to the bivariant E-theory of Connes and
Higson. With this background, we review how Weyl--Moyal quantization may be
described using the tangent groupoid. Subsequently, we explain how the
Baum--Connes analytic assembly map in E-theory may be seen as an equivariant
version of Weyl--Moyal quantization. Finally, we expose Connes's tangent
groupoid proof of the Atiyah--Singer index theoremComment: 16 pages, Proc. Constanta 200