I will sketch the proof that every connected affine scheme in positive characteristic is a K(pi,1) space for the etale topology. The key technical ingredient is a “Bertini-type” statement regarding the wild ramification of l-adic local systems on affine spaces. Its proof uses in an essential way recent advances in higher ramification theory due to T. Saito. Time permitting, I will discuss some "anabelian" and "irregular" ramifications of the result.