This is the second of a series of 5 volumes devoted to spectral asymptotics for partial differential equations, from the perspective of the semiclassical microlocal analysis. We address to vol. 1 and respective review for a general description of the contents of all the volumes, mainly concerning the contributions of the author Victor Ivrii to the field. The results of the present vol. 2 can be seen as applications of those in vol. 1, hence the preliminary reading of vol. 1 is essential. Even, numbers of chapters and formulas are continuations of those in vol. 1, and Bibliograpy is the same. On the other hand, classical functional methods are also needed in this new part, and in such respect the book reflects the educational background of the author and links with the Russian School, in particular with papers of M. Sh. Birman and M. Z. Solomiak and their students G. Rozenblioum, Yu. Safarov, A. Sobolev. The initial Chapter 9, devoted to the negative spectrum of an operator semibounded from below, is representative of this approach. In fact, in the study of domains with angles, spikes and cusps, semiclassical asymptotics are applied in the regular zone, whereas in the singular zone the variational estimates of Rozenblioum are used, cf. the contributions of Ivrii 1986. Same point of view is adopted in Chapter 10, concerning estimates of the spectrum in an interval. Here the Birman-Schwinger principle is applied, and this allows to reduce asymptotics with respect to the spectral parameter to asymptotics with respect to the semiclassical parameter (Ivrii 1987). Chapter 11 and 12, devoted to Weyl asymptotics of spectra, are the core of the volume. In particular Chapter 11 gives the celebrated results of Ivrii 1985–86 concerning remainder estimates, and new extensions. The conclusive Chapter 12 represents a tremendous effort of the author to reconsider several results in literature from his point of view, by giving an improved version with sharp remainders. Let us list in short the concerned classes of operators: operators in domains with thick cusps, operators with degenerate potential growing at infinity or singular potential decaying at infinity, maximally hypoelliptic operators with singularities, domains with thin cusps and periodic operators. The reader may find there an encyclopedia of results on spectral asymptotics, presented with sharp remainder estimates.