  We develop a quantum harmonic analysis framework for the affine group. This
encapsulates several examples in the literature such as affine localization
operators, covariant integral quantizations, and affine quadratic
time-frequency representations. In the process, we develop a notion of
admissibility for operators and extend well known results to the operator
setting. A major theme of the paper is the interaction between operator
convolutions, affine Weyl quantization, and admissibility.
