Abstract: It is a problem of great interest to determine which simply connected, closed manifolds admit metrics of non-negative sectional curvature. Considering Gromov's Betti number bound there remains a wide gap between known examples and obstructions. We present a new method of construction, originally suggested by B.\ Wilking, for non-negative sectional curvature and use it to construct such metrics on a family of 2-connected, 7-manifolds. Specifically we show: there are many 7-manifolds homeomorphic to, but not necessarily diffeomorphic to $\mathbf{S}^3$-bundles over $\mathbf{S}^4$ that admit non-negative sectional curvature. In particular, all 28 smooth structures on the sphere $\mathbf{S}^7$ admit such metrics. This completes the picture for exotic 7-spheres, where such metrics were known only for the Milnor spheres (those that are diffeomorphic to $\mathbf{S}^3$-bundles over $\mathbf{S}^4$) due to Gromoll--Meyer and Grove--Ziller. This is joint work with Sebastian Goette and Martin Kerin.