  Starting from Borel's description of the mod-2 cohomology of real flag
manifolds, we give a minimal presentation of the cohomology ring for semi
complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$ where $1$ is repeated $k$
times. The information is used in order to estimate Farber's topological
complexity of these spaces when $m$ approaches (from below) a 2-power. In
particular, we get almost sharp estimates for $F_{2,2^e-1}$ which resemble the
known situation for the real projective spaces $F_{1,2^e}$. Our results
indicate that the agreement between the topological complexity and the
immersion dimension of real projective spaces no longer holds for other flag
manifolds. More interestingly, we also get corresponding results for the $s$-th
(higher) topological complexity of these spaces. Actually, we prove the
surprising fact that, as $s$ increases, the estimates become stronger. Indeed,
we get several full computations of the higher motion planning problem of these
manifolds. This property is also shown to hold for surfaces: we get a complete
computation of the higher topological complexity of all closed surfaces
(orientable or not). A homotopy-obstruction explanation is included for the
phenomenon of having a cohomologically accessible higher topological complexity
even when the regular topological complexity is not so accessible.
