I will review the Kostant-Souriau geometric quantization procedure for
passing from functions on a symplectic manifold (classical observables)
to operators on a Hilbert space (quantum observables).
With the "half-form correction" that is required in this procedure,
one cannot quantize a complex projective space of even complex dimension,
and one cannot equivariantly quantize the two-sphere nor any symplectic
toric manifold.
I will present a geometric quantization procedure that uses metaplectic-c
structures to incorporate the half-form correction into the earlier
prequantization stage and that does apply to these examples.
This follows work of Harald Hess from the late 1970s with recent contributions of Jennifer Vaughan.