Curvature cones and the Ricci flow.

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters.