I will report on recent results concerning the anabelian geometry of hyperbolic curves over finite fields due to S. Mochizuki, A. Tamagawa and myself. This includes a prime to p-version of the anabelian Grothendieck conjecture for hyperbolic curves over finite fields, and a Hom-form of the birational Grothendieck conjecture for function fields (under suitable local conditions) that were both recently proven by A. Tamagawa and myself in a joint work.