The two most important polymorphs of aluminum hydroxide, namely gibbsite and bayerite, have been studied for the first time using solid state ab initio quantum mechanical methods, both using plane wave and localized basis set methodologies, within the framework of nonlocal density functional theory. The fully optimized structures have been determined for both phases, yielding improved hydrogen positions in the case of gibbsite for which the only previous information is from X-ray data. Mechanical properties have been calculated for gibbsite, including the full elastic constants tensor and the bulk modulus. The latter is found to be 55 GPa, which is significantly lower than a recent experimental estimate. Vibrational spectra have been calculated for both phases and assignments of the hydroxyl stretching modes are proposed.