The first part of the talk will be an introduction to the index/K-theory approach to quantization. Then, depending on time, I will introduce Hamiltonian loop-group spaces (which have a close connection to moduli spaces of flat connections on Riemann surfaces), and describe some work in progress with E. Meinrenken and Y. Song, in which we study a 'quantization' of these spaces using a technique called norm-square localization.