Hydroxyapatite (HAp) is an important component of mammal bones and teeth, being widely used in prosthetic implants. Despite the importance of HAp in medicine, several promising applications involving this material e.g. in photo-catalysis), depend on how well we understand its fundamental properties... Among the ones that are either unknown or not known accurately we have the electronic band structure and all that relates to it, including the band gap width. We employ state-of-the-art methodologies, including density hybrid-functional theory and many-body perturbation theory within the GW approximation, to look at the optoelectronic properties of HAp. These methods are also applied to the calculation of defect levels. We find that the use of a mix of (semi-)local and exact exchange in the exchange-correlation functional, brings a drastic improvement to the band structure. Important side-effects include improvements in the description of dielectric and optical properties, not only involving conduction band (excited) states, but also the valence. We find that the highly dispersive conduction band bottom of HAp originates from anti-bonding $\sigma^{*}$ states along the $\cdots\textrm{OH-OH-}\cdots$ infinite chain, suggesting the formation of a conductive 1D-ice phase. The choice of the exchange-correlation treatment to the calculation of defect levels was also investigated by using the OH-vacancy as testing-model. We find that donor and acceptor transitions obtained within semi-local DFT differ from those of hybrid-DFT by almost 2 eV. Such a large discrepancy emphasizes the importance of using a high-quality description of the electron-electron interactions in the calculation of electronic and optical transitions of defects in HAp. (read more)