It took us a long time to write this book. In 1959, two of us (Lifshits and Kagav) pub lished a review of the mechanics of electrons with a complex dispersion law. About that time, geometrical terms such as extremal sections, curvatures, diameters, limiting points began to appear in papers on the electron theory of metals. They were followed by terms quite unusual in the scientific literature: monsters, pockets, arms, sheets, and so on. With their excitingly shaped figures, papers on the electron theory of metals began to resemble catalogs of exhibitions of abstract or ultramodern sculpture. The modern theory of metals was passing through its romantic period. Each newly interpreted Fermi surface and each discovery of a new structure sensitive phemen was an emotional experience for the authors and readers alike. The atti tude of the theoreticians was epitomized by phrases such as This method or this phemen can be used to reconstruct the Fermi surface . . ., which were found at the end of almost every paper on the electron theory of metals. The experimentalists selected convenient meth ods, being guided t so much by the elegance of a particular method as by its experimental capabilities. Gradually, the romantic approach gave way to a systematic activity, which re sulted in the interpretation of the energy spectra of the majority of metals. There were some unavoidable disappointments.