This book makes a systematic study of the relations between the ?tale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, ?tale cohomology is cohomology of the real spectrum. It also contains new contributions in group cohomology and in topos theory. It is of interest to graduate students and researchers who work in algebraic geometry (not only real) and have some familiarity with the basics of ?tale cohomology and Grothendieck sites. Independently, it is of interest to people working in the cohomology theory of groups or in topos theory.