We show that Kervaire invariant one elements in the homotopy groups of
spheres exist only in dimensions at most 126. By Browder's Theorem, this means
that smooth framed manifolds of Kervaire invariant one exist only in dimensions
2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this
resolves a longstanding problem in algebraic topology.