ROSAT Observations G. HASINGER Max-Planck-Institut flir extraterrestrische Physik, D-85740 Garching, Germany Abstract. This review describes the most recent advances in the study of the extragalactic soft X-ray background and what we can learn about its constituents. The deepest pointed observations with the ROSAT PSPC are discussed. The logN-logS relation is presented, which reaches to the faintest X-ray fluxes and to the highest AGN surface densities ever achieved. The N(>S) relation shows a 2 density in excess of 400 deg- at the faintest fluxes and a flattening below the Einstein Deep...