I will begin by recalling and discussing basic properties of various invariants associated to an arbitrary projective variety, that aim to measure how far it is from being rational, rationally connected or uniruled respectively. In the second part of the talk I will discuss recent work with Kouvidakis computing these invariants for the Fano scheme of lines of a cubic threefold, which is a general type surface classically studied by Fano, Clemens-Griffiths and others.