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

<p>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=<b>C</b>,
this brings nothing really new.

<p>Over fields k which are finitely generated over <b>Q</b>, 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
<i>anabelian</i>, expecting that their fundamental groups would be "far
away from being abelian''. In the 1990s, Tamagawa and Mochizuki proved
that hyperbolic curves are anabelian.

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