High Resolution X-ray Spectroscopy of Galaxies and Clusters with AXAF

We review the high resolution X-ray spectroscopy capabilities of the AXAF observatory focusing on the High Energy Transmission Grating (HETG). As part of the Guaranteed Time Observation (GTO) program, the HETG science team will observe both elliptical galaxies and clusters of galaxies. We discuss the problems associated with observing extended sources with the HETG and some of the potential scientific insights which AXAF's spectroscopy capabilities can provide for these classes of objects.