The goal of Grothendieck's anabelian geometry is to reconstruct 
the geometry of schemes in terms of profinite groups, via their 
etale fundamental groups. In the case of curves over finite 
fields, the fundamental conjecture in anabelian geometry 
has been proved, for function fields in the mid-1970's (Uchida), 
for affine curves in the mid-1990's (Tamagawa), and for proper 
curves in the mid-2000's (Mochizuki). In this talk, we shall 
discuss a recent generalization of these results, where the 
profinite fundamental groups are replaced by various quotients, 
such as geometrically pro-$\Sigma$ fundamental groups for certain 
infinite sets $\Sigma$ of primes.