  We develop a finite KKG-theory of C*-algebras following Arlettaz-
H.Inassaridze's approach to finite algebraic K-theory. The Browder-
Karoubi-Lambre's theorem on the orders of the elements for finite algebraic
K-theory is extended to finite KKG-theory. A new bivariant theory, called
torsion KK-theory is defined as the direct limit of finite KK-theories. Such
bivariant K-theory has almost all KKG-theory properties and one has a long
exact sequence relating KK-theory, rational bivariant K-theory and torsion
KK-theory. For a given homology theory on the category of separable
GC*-algebras finite, rational and torsion homology theories are introduced and
investigated. In particular, we formulate finite, torsion and rational versions
of Baum-Connes Conjecture. The later is equivalent to the investigation of
rational and q-finite analogues for Baum-Connes Conjecture for all prime q.
