The fast and autonomous response of the Swift satellite to newly detected Gamma-Ray Bursts (GRBs) has provided a new view on their X-ray and optical afterglows. We present here the X-ray Telescope observations of GRB 050607. The striking feature of the X-ray emission is the intense flaring activity during the early afterglow phase, indicative of central engine activity extended to several hundred seconds after the burst detection. The flares have very rapid timing variations and asymmetric shapes, similar to the FREDs that are frequently observed in the prompt GRB emission. After the flares, the X-ray lightcurve entered a phase of slow decay during which the forward shocks were probably being refreshed, followed by a late steepening consistent with the standard external shock model. Analysis of the X-ray emission showed spectral variations during the flares, with a harder spectrum at the flares onset and a softer component that lingered longer, as each flare decayed.