The Swift Gamma Ray Burst Explorer will be launched in 2003 to observe hundreds of gamma-ray bursts per year and study their X-ray and optical afterglows, using a multiwavelength complement of three instruments: a wide-field Burst Alert Telescope (BAT), an X-Ray Telescope (XRT), and a UV/Optical Telescope (UVOT). The XRT is designed to study X-ray counterparts of the gamma-ray bursts and their afterglows, beginning 20 - 70 s from the time of the burst, and continuing for days or weeks. The XRT utilizes a superb mirror set built for JET-X and a state-of-the-art XMM/EPIC CCD detector to provide a sensitive broad-band (0.2 - 10 keV) X-ray imager with effective area of 110 cm2 at 1.5 keV, field of view of 23.6 X 23.6 arcminutes, and angular resolution of 15 arcseconds (HEW). The sensitivity is 2 X 10-14 erg/cm2s in 104 seconds. The telescope electronics are designed to provide automated source detection and position reporting, with a position good to 2.5 arcseconds transmitted to the ground within 100 seconds of the burst detection. The XRT will operate in an auto-exposure mode, adjusting the CCD readout mode automatically to optimize the science return for each frame as the source fades. The XRT will measure spectra and lightcurves of the GRB afterglow beginning within about a minute after the burst and will follow each burst until it fades from view, typically monitoring 2 - 3 'old' bursts at a time while waiting for a new burst to be detected.