I. Picard-Lefschetz-Pham theory and singularity theory -- § 1. Gauss-Manin connection in the homological bundles. Monodromy and variation operators -- § 2. The Picard-Lefschetz formula. The Leray tube operator -- § 3. Local monodromy of isolated singularities of holomorphic functions -- § 4. Intersection form and complex conjugation in the vanishing homology of real singularities in two variables -- § 5. Classification of real and complex singularities of functions -- § 6. Lyashko-Looijenga covering and its generalizations -- § 7. Complements of discriminants of real simple singularities (after E. Looijenga) -- § 8. Stratifications. Semialgebraic, semianalytic and subanalytic sets -- § 9. Pham's formulae -- § 10. Monodromy of hyperplane sections -- § 11. Stabilization of local monodromy and variation of hyperplane sections close to strata of positive dimension (stratified Picard-Lefschetz theory) -- § 12. Homology of local systems. Twisted Picard-Lefschetz formulae -- § 13. Singularities of complete intersections and their local monodromy groups -- II. Newton's theorem on the nonintegrability of ovals -- § 1. Stating the problems and the main results -- § 2. Reduction of the integrability problem to the (generalized) PicardLefschetz theory -- § 3. The element "cap" -- § 4. Ramification of integration cycles close to nonsingular points. Generating functions and generating families of smooth hypersurfaces -- § 5. Obstructions to integrability arising from the cuspidal edges. Proof of Theorem 1.8 -- § 6. Obstructions to integrability arising from the asymptotic hyperplanes. Proof of Theorem 1.9 -- § 7. Several open problems -- III. Newton's potential of algebraic layers -- § 1. Theorems of Newton and Ivory -- § 2. Potentials of hyperbolic layers are polynomial in the hyperbolicity domains (after Arnold and Givental) -- § 3. Proofs of Main Theorems 1 and 2 -- § 4. Description of the small monodromy group -- § 5. Proof of Main Theorem 3 -- IV. Lacunas and the local Petrovski$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{I} $$ condition for hyperbolic differential operators with constant coefficients -- § 0. Hyperbolic polynomials -- § 1. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion and lacunas -- § 2. Generating functions and generating families of wave fronts for hyperbolic operators with constant coefficients. Classification of the singular points of wave fronts -- § 3. Local lacunas close to nonsingular points of fronts and to singularities A2, A3 (after Davydova, Borovikov and Gárding) -- § 4. Petrovskii and Leray cycles. The Herglotz-Petrovskii-Leray formula and the Petrovskii condition for global lacunas -- § 5. Local Petrovskii condition and local Petrovskii cycle. The local Petrovskii condition implies sharpness (after Atiyah, Bott and Gárding) -- § 6. Sharpness implies the local Petrovskii condition close to discrete-type points of wave fronts of strictly hyperbolic operators -- § 7. The local Petrovskii condition may be stronger than the sharpness close to singular points not of discrete type -- § 8. Normal forms of nonsharpness close to singularities of wave fronts (after A.N. Varchenko) -- § 9. Several problems -- V. Calculation of local Petrovski$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{I} $$ cycles and enumeration of local lacunas close to real function singularities -- § 1. Main theorems -- § 2. Local lacunas close to singularities from the classification tables -- § 3. Calculation of the even local Petrovskii class -- § 4. Calculation of the odd local Petrovskii class -- § 5. Stabilization of the local Petrovskii classes. Proof of Theorem 1.5 -- § 6. Local lacunas close to simple singularities -- § 7. Geometrical criterion for sharpness close to simple singularities -- § 8. A program for counting topologically different morsifications of a real singularity -- § 9. More detailed description of the algorithm -- Appendix: a FORTRAN program searching for the lacunas and enumerating the morsifications of real function singularities.