The evolution of the Martian atmosphere, particularly with regard to water, is influenced by (1) ''nonthermal'' escape of oxygen atoms created by dissociative recombination and (2) by oxygen ion pickup by the solar wind. Both processes depend on the intensity of solar EUV radiation, which affects atmosphere temperatures (scale heights) and photoionization rates, and thereby the exosphere and the fluxes of escaping atoms and ions. This study involves the calculation, by the two-stream model method of Nagy and Cravens (1988), of the exospheric hot oxygen densities for ''ancient'' atmospheres and ionospheres (e.g., for different EUV fluxes), and the associated neutral escape fluxes. The ion production rates above nominal ionopause altitudes are also computed and are considered to be the upper limits to losses by direct solar wind pickup. Since we do not consider the pickup ion precipitation and additional neutral escape due to the sputtering process described by Luhmann and Kozyra (1991), the results presented here represent conservative estimates of the neutral escape fluxes, but somewhat generous estimates of ion loss rates. We find that when the inferred increased solar EUV fluxes of the past are taken into account, oxygen equivalent to that in several tens of meters of water, planet-wide, should have escaped to space over the last 3 Gyr. ¿ American Geophysical Union 1993