The electronic and local structural properties of CuO under pressure have been investigated by means of X-ray absorption spectroscopy (XAS) at Cu K edge and ab initio calculations, up to 17 GPa. The crystal structure of CuO consists of Cu motifs within CuO4 square planar units and two elongated apical Cu-O bonds. The CuO4 square planar units are stable in the studied pressure range, with Cu-O distances that are approximately constant up to 5 GPa, and then decrease slightly up to 17 GPa. In contrast, the elongated Cu-O apical distances decrease continuously with pressure in the studied range. An anomalous increase of the mean square relative displacement (EXAFS Debye-Waller, σ2) of the elongated Cu-O path is observed from 5 GPa up to 13 GPa, when a drastic reduction takes place in σ2. This is interpreted in terms of local dynamic disorder along the apical Cu-O path. At higher pressures (P > 13 GPa), the local structure of Cu2+ changes from a 4-fold square planar to a 4+2 Jahn-Teller distorted octahedral ion. We interpret these results in terms of the tendency of the Cu2+ ion to form favorable interactions with the apical O atoms. Also, the decrease in Cu-O apical distance caused by compression softens the normal mode associated with the out-of-plane Cu movement. CuO is predicted to have an anomalous rise in permittivity with pressure as well as modest piezoelectricity in the 5-13 GPa pressure range. In addition, the near edge features in our XAS experiment show a discontinuity and a change of tendency at 5 GPa. For P < 5 GPa the evolution of the edge shoulder is ascribed to purely electronic effects which also affect the charge transfer integral. This is linked to a charge migration from the Cu to O, but also to an increase of the energy band gap, which show a change of tendency occurring also at 5 GPa.