In topology the Eilenberg MacLane spaces K(\pi,1) are precisely those spaces, where the fundamental group controls everything. In anabelian geometry, we hope to find analogues of K(\pi, 1) spaces but in the algebraic category. The talk will first introduce a cohomological condition for a space to be a pro-finite K(\pi,1) space, that essentially goes back to Artin-Mazur's higher algebraic homotopy theory. Then, I will discuss several geometric consequences for smooth, proper varieties that are K(\pi,1)'s and conclude with an example from number theory.