Fibrations of the round sphere by great subspheres entered the collective mathematical consciousness through a connection to the Blaschke Problem (the Blaschke Problem asks whether a space, all of whose geodesics (starting from any point) stop being minimizing after the same distance, must be a sphere or projective space with its standard metric). The prototypical great- subsphere fibrations are the Hopf fibrations, and a natural question is whether they have any special features which uniquely characterize them among great-subsphere fibrations. I´ll discuss recent work of mine in which I prove that the Hopf fibrations are characterized by being fiberwise homogeneous: for any pair of fibers, there is an isometry of the total space which takes fibers to fibers and takes the first given fiber to the second.