Abstract. With the successful launch of AXAF next summer, it will be possible to observe the X-ray afterglow from gamma-ray bursts with much improved positional accuracy (∼ 1 arcsec) and sensitivity (2 − 10 keV, 1.5 10 −14 ergs/cm 2 /s in 10 ksec). A typical burst afterglow should be observable within about 10 hours after the position has been determined by BeppoSAX or any other satellite capable of determining the position to within about 5 arcmin and provide about 1000 counts in a 10 ksec integration using the ACIS on AXAF. To constrain the decrease of intensity with time, using existing data from BeppoSAX and ASCA, the burst should produce a measurable flux for up to about 4 months after the gamma-ray event for an observing time of about 40 ksec, if a simple extrapolation of the observed fall-off with time is assumed. Depending upon a number of assumptions, there is a slight possibility of observing iron lines from the gas immediately surrounding the gamma-ray burst source using a long exposure (> 100 ksec).