Dispersive spectroscopy on AXAFThe designs of two transmission grating spectrometers and a Bragg crystal spectrometer that are being developed for the Advanced X-ray Astrophysics Facility (AXAF) are described. The gratings, which are composed of arrays of small facets mounted on plates which can be inserted immediately behind the AXAF telescope, divide the AXAF energy band (80 eV-10 keV) into three regions and attain very high resolving powers for point sources. The Bragg Crystal Spectrometer (BCS) is a focal plane instrument, in which X-rays that satisfy the Bragg law are reflected from a curved crystal which refocuses the beam onto an imaging detector. The BCS achieves the highest spectral resolutions of the AXAF spectrometers: for E values between 500 and 1600 eV, the FWHM of a narrow line is less than 1 eV.