As Ryan Budney indicated, Mark Behrens gave a talk about joint work with Mike Hill, Mike Hopkins, and Mark Mahowald at the Milnor conference a few days ago. Here's a link to the video.
The "reader's digest" version of this is that, by using periodic families of elements in the stable homotopy groups of spheres, they've established a large number of congruences where exotic spheres are always forced to exist, but it's not yet exhaustive. In addition, they've done enough low-dimensional computations to know that in dimensions less than or equal to (at least) 63, there are exotic spheres in all dimensions except 1, 2, 3, 4, 5, 6, 12, and 61. The first of these were known from work of Kervaire-Milnor in 1963 (as part of their enumeration of the number of exotic spheres that's indicated by the Wikipedia entry), but the nonexistence of exotic spheres in dimension 61 is a new result.
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