In response to a question of Milnor, a collaboration of Mike Hill, Mike Hopkins, Mark Mahowald, and myself have been attempting to determine in which dimensions exotic spheres exist. We will review the work of Kervaire-Milnor, which reduces the problem to questions in the stable homotopy groups of spheres. Thanks to the solution of the Kervaire invariant one problem, the problem is reduced to producing certain infinite families in the stable stems. I will discuss what we know, and don't know about the stable stems these days, and discuss some joint work with Mark Mahowald, producing a plethora of 192-periodic families (the Hurewicz image of TMF) which give partial answers to the original question.