This talk will revolve around the following question: If M is a non-negatively curved manifold, can we deform the metric to achieve positive curvature? Among other things we will see: - that the set of footpoints of planes of zero curvature plays a major role in determining the answer; - why this is not a good approach to the Hopf conjecture; - what the deformation conjecture is, why it is false, and why the Ricci and scalar curvature analogs of it are true; - how this has produced some of the (few) examples of manifolds with positive curvature, through so-called Cheeger deformations.