The structure, electronic properties and relative stability of seven thermodynamically stable, meta-stable and high-pressure alumina polymorphs as well as the structure and relative stability of four aluminum hydroxides were calculated with periodic hybrid density functional theory calculations and compared with available experimental data. For a number of polymorphs several structure models that are discussed in the literature were compared in terms of their agreement with structural data and stability. In order to compare oxides and hydroxides the energies and heats of atomization of the latter were corrected by the reaction energy with water. The following overall energetic order was obtained: gibbsite < bayerite < boehmite < akdalaite < [small alpha]-Al2O3 < [small kappa]-Al2O3 < [small theta]-Al2O3 < [small delta]-Al2O3 < [gamma]-Al2O3 < [small eta]-Al2O3 < [iota]-Al2O3.