Sorting networks are shortest paths between the identity and the reverse permutation in the Cayley graph of the symmetric group generated by adjacent transpositions. Some remarkable conjectures have been made on the geometric behavior of uniform random sorting networks. For example, the half way permutation appears to be the projection of spherical measure onto the disk. I will explain these conjectures, its relation to the limit theory of permutations and some recent progress supporting these conjectures in joint work with Balint Virag and Mate Vizer.