Zirconium and hafnium oxide films are being extensively studied asalternative gate dielectrics for field effect transistors. Dielectric properties of currently produced films strongly depend on deposition techniques, suggesting importance of electronic defects in these materials. In particular, shallow electron traps whose occupancy may change under small variations of external potential, are likely to contribute to the leakage currents. In this paper we investigate the r\`{o}le of oxygen and hydrogen defects on degradation of dielectric properties of monoclinic zirconia and hafnia. We apply plane wave density functional theory to model structure, ionisation energies and electron affinities of the oxygen vacancy and interstitial defects as well as hydrogen contained defects in different charge states. Since unambiguous characterisation of hydrogen defects can only be achieved by means of infrared spectroscopy, we use {\it ab initio} molecular dynamics to calculate vibrational properties and diffusion mechanisms of water and hydrogen species in $ZrO_2$. We discuss the properties of the defects studied in light of their possible contribution to the leakage currents and films degradation. The question of reliability of the density functional methods for calculations of shallow trap centers in oxides will also be addressed.