This paper is a survey article which outlines the known results about the existence of non-negative sectional curvature metrics on disc bundles. The search for manifolds of nonnegative curvature is a classical problem in Riemannian geometry; however, there are few general classes of examples and construction methods. Apart from taking products, there are only two general methods to construct new non-negatively curved metrics out of given manifolds. While one of the methods uses Riemannian submersions which do not decrease curvature, the other is based on the gluing of two manifolds along their common boundary. As a consequence, one of the principal construction methods of closed manifolds without boundary with non-negative sectional curvature is to glue together a pair of non-negatively curved disc bundles with isometric totally geodesic boundaries. But this process can be fulfilled only in special situations. In this paper, the author surveys a series of examples and recent results which show the difficulty in finding metrics on disk bundles which are suitable for the gluing construction method.
For the entire collection see [Zbl 1230.53006].