There are few known obstructions for positive sectional curvature, yet the class of manifolds with non-negative curvature is huge compared to the known examples of manifolds with positive sectional curvature. The Hopf conjecture inquires as to whether M = S^2 x S^2 has a metric of positive curvature, where the product metric clearly endows M with non-negative curvature. Gromoll and Meyer conjectured that a manifold of non-negative curvature, which has a point at which all sectional curvatures are positive, admits a metric of positive curvature. The analogous statements for Ricci and scalar curvature are true. We present some of Wilking's counterexamples to the Gromoll and Meyer conjecture, and in so doing present an argument for a class of manifolds between those which admit positive and non-negative curvature respectively, namely those which admit positive curvature almost everywhere.