Abstract:Two years ago in a talk at the University of Toronto, I discussed certain similarities between an equivariant index formula for contact manifolds that became part of my thesis, and the equivariant Riemann-Roch formula, which appears in the reformulation of Souriau-Kostant geometric quantization as an equivariant index. A natural question then follows: is there a contact version of geometric quantization to which this contact index formula corresponds? I will explain my attempt to answer this question in the affirmative. In particular, I will explain what I believe to be the correct contact analogues of the algebra of observables, the prequantum line bundle, and polarization. In the special case of a regular contact manifold, this approach reduces to the Souriau-Kostant construction: the quotient of the contact quantum bundle by the Reeb flow gives a prequantum line bundle over a symplectic manifold.