  We give an algorithm to compute the integer cohomology groups of any real
partial flag manifold, by computing the incidence coefficients of the Schubert
cells. For even flag manifolds we determine the integer cohomology groups, by
proving that any torsion class has order 2 (generalizing a result of
Ehresmann). We conjecture this to hold for any real flag manifold. We obtain
results concerning which Schubert varieties represent integer cohomology
classes, their structure constants and how to express them in terms of
characteristic classes. For even flag manifolds and Grassmannians we also
describe Schubert calculus. The Schubert calculus can be used to obtain lower
bounds for certain real enumerative geometry problems (Schubert problems).
