ROSAT observations of globular clusters in the galaxy and in M31

X-ray sources in globular clusters fall into two categories: the ''bright'' sources, with L(X) similar to 10(36)-10(38) ergs(-1), and the ''dim'' sources, with L(X) less than or similar to 10(34.5) ergs(-1). The bright sources are clearly associated with accreting neutron stars in binary systems. The nature of the dim sources, however, remains in doubt. We review recent observations of globular-cluster X-ray sources with the ROSAT satellite. ROSAT detected bright sources in M31 globular clusters and greatly increased the number of dim sources known in galactic globular clusters. We discuss what these new observations have taught us about the distribution and nature of such sources, their spectral properties, and their underlying luminosity function.