  We define an analytical index map and a topological index map for conical
pseudomanifolds. These constructions generalize the analogous constructions
used by Atiyah and Singer in the proof of their topological index theorem for a
smooth, compact manifold $M$. A main ingredient is a non-commutative algebra
that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism
between non-commutative algebras which gives a new example of wrong way
functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer
index theorem using deformation groupoids and show how it generalizes to
conical pseudomanifolds. We thus prove a topological index theorem for conical
pseudomanifolds.
