Abstract: For a complex algebraic variety X, the underlying topological space X(C) gives only limited information on the algebraic structure of X (e.g. the topological spaces E(C) of elliptic curves are all homeomorphic).

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.