  We report properties of the reported transparent conductor CuI, including the
effect of heavy $p$-type doping. The results, based on first principles
calculations, include analysis of the electronic structure and calculations of
optical and dielectric properties. We find that the origin of the favorable
transparent conducting behavior lies in the absence in the visible of strong
interband transitions between deeper valence bands and states at the valence
band maximum that become empty with $p$-type doping. Instead, strong interband
transitions to the valence band maximum are concentrated in the infrared with
energies below 1.3 eV. This is contrast to the valence bands of many wide band
gap materials. Turning to the mobility we find that the states at the valence
band maximum are relatively dispersive. This originates from their antibonding
Cu $d$ - I $p$ character. We find a modest enhancement of the Born effective
charges relative to nominal values, leading to a dielectric constant
$\varepsilon(0)$=6.3. This is sufficiently large to reduce ionized impurity
scattering, leading to the expectation that the properties of CuI can be still
can be significantly improved through sample quality.
