We report a comprehensive study of the defects in room-temperature ferromagnetic (RTFM) Cu-doped ZnO thin films using x-ray absorption spectroscopy. The films are doped with 2 at.% Cu, and are prepared by reactive magnetron sputtering (RMS) and pulsed laser deposition (PLD), respectively. The results reveal unambiguously that atomic point defects exist in these RTFM thin films. The valence states of the Cu ions in both films are 2(+). In the film prepared by PLD, the oxygen vacancies (V(O)) form around both Zn ions and Cu ions in the hexagonal wurtzite structure. Upon annealing of the film in O(2), the V(O) population reduces and so does the RTFM. In the film prepared by RMS, the V(O)s around Cu ions are not detected, and the V(O) population around Zn ions is also smaller than in the PLD-prepared film. However, zinc vacancies (V(Zn)) are evidenced. Given the low doping level of spin-carrying Cu ions, these results provide strong support for defect-mediated ferromagnetism in Cu-doped ZnO thin films.