  A consistent exposition of the arguments and constructions of the method of
maximal singularities, the aim of which is to describe birational
iso/automorphisms of Fano varieties and Fano fibrations. The principal elements
of the method are considered: N{\" o}ther-Fano inequality, maximal cycles,
infinitely near maximal singularities, exclusion and untwisting. In a detailed
way the crucial technical points are discussed. We also give a new version of
the proof of Sarkisov theorem which is ideologically more close to the original
arguments of V.A.Iskovskikh and Yu.I.Manin.
