Classical Baum-Connes conjecture formulated for group actions,
has attracted vigorous research in the last two decades
and several areas of mathematics, notably K-theory, topology,
geometry, and theory of operator algebras have been further
developed and enriched through these research efforts.

In this talk, we present a formulation of the conjecture
for the action of discrete quantum groups and test the conjecture
with some examples. This new formulation is a generalization
of the earlier formulation for discrete groups with prospects
for further applications.

The talk starts with a definition of the basic notions
involved in the classical formulation of the conjecture - full
and reduced group C*-algebras, K-theory and K-homology for C*-algebras,
proper group actions, universal space for proper actions,
assembly maps - leading to the classical formulation of the conjecture.
We also briefly review the state of knowledge on the conjecture.

Next, we define quantum analogues of the notions above - discrete
quantum group, A, say; equivariant and proper A-actions,
A-equivariant K-theory and K-homology, new assembly maps etc. culminating
in the new formulation of the conjecture. Finally we show that
the new conjecture holds for finite dimensional quantum groups
and prove subjectivity of the new assembly map for the dual of quantum SU(2).