Manifolds with non-negative sectional curvature are little understood, so any new topological types are treated with great interest. In this talk, based upon joint work with S. Goette and K. Shankar, I will exploit the linking form to show that there are infinite families of 2-connected, rational homology 7-spheres which admit non-negative curvature and are not even homotopy equivalent to an S^3-bundle over S^4. These are the first such examples.