Rational Connectivity

A variety is rationally connected if two general points can be connected by a rational curve. By way of example, I shall briefly sketch why Fano varieties, with ample anticanonical class, are rationally connected. I shall then illustrate their utility by explaining an argument of Graber, Harris, and Starr that a fibration over a curve has a section if the general fibre is rationally connected. I hope to at the end sketch some natural further generalisations involving an analogy with homotopy theory.