  In this short review we pass through the milestones in the studies of the
electronic topological transitions (ETT) and focus on some recent applications
of the ideas worked out in their classical theory. These are: two-dimensional
electron systems, de Haas-van Alphen effect, classification of ETT in
multidimensional systems, superconductivity in systems close to ETT,
thermoelectricity in heavy-fermion systems, where the cascades of topological
changes of Fermi surface (FS) are generated by magnetic field. The history of
studies of ETT is inextricably linked with Kharkov school of condensed matter
physics, with such names as I.M. Lifshitz, V.G.Bar'yakhtar and many other.
Among them is Moisey Isaakovich Kaganov, who contributed much in studies of the
role of geometry and topology of FS in physical properties of the metals. Two
of the authors (A.V. and Y.G.) had a honor and pleasure to work with "Musik",
as all friends called Kaganov; all of us have been learning the niceties of
science from his books. "The Fermi surface is the stage on which the drama of
the life of the electron is played out" wrote Kaganov and Lifshitz. We devote
this work to their memory.
