This paper is devoted to the proof of the following conjecture (due to Fano and Iskovskikh): let \(V\) be a Fano threefold with canonical Gorenstein singularities, then \(-K_X^3 \leq 72\). Moreover, if the equality holds, \(V\) is shown to be isomorphic to a weighted projective space (either \({\mathbb P}(3,1,1,1)\) or \({\mathbb P}(6,4,1,1)\)).
The proof is based on the minimal model program and on the properties of a weak Fano threefold, i.e. a threefold whose anticanonical divisor is a nef and big \({\mathbb Q}\)-Cartier divisor. More precisely, by means of a suitable modification \(\phi: W \to V\), the author reduces the statement to a similar one for a weak Fano threefold \(W\) with terminal factorial singularities. Then the required result follows from a detailed analysis of the structure of the extremal rays of \(W\).