For varieties over a general field k, the set of points X(k) does
not carry a reasonable topology, but there is a more sophisticated
construction assigning to X a topological space X_{et}. For k=**C**,
this brings nothing really new.

Over fields k which are finitely generated over **Q**, however, the
situation seems to be rather rigid. Grothendieck predicted the
existence of a class of varieties which are reconstructible up to
isomorphism from the fundamental group of X_{et}. He called them
*anabelian*, expecting that their fundamental groups would be "far
away from being abelian''. In the 1990s, Tamagawa and Mochizuki proved
that hyperbolic curves are anabelian.

In this talk we explain the setting and also present new results. This is joint work with J. Stix.