1 Introduction. - 2 Classical theory of nonlinear vibrational systems; local modes. - 2. 1 Survey of classical mechanics; introduction to local modes. - 2. 2 Method of averaging. - 2. 3 Two weakly coupled Morse oscillators: dynamics and local - mode formation. - 2. 4 Analysis of phase trajectories with the help of Poincare sections numerical results. - 3 Coupled nonlinear oscillators: formation and decay of local modes. - 3. 1 Two weakly coupled nonlinear oscillators: decay of normal modes. - 3. 2 Multi-oscillator systems: local modes' and normal modes' representations. - 3. 3 Morse oscillators: calculation of resonance linewidths and criteria for the appearance of local modes. - 3. 4 Coupled nonlinear oscillators: localized vibrational excitations and their asymptotic stability. - 3. 5 Canonical transformation to local - mode representation. - 4 Discrete breathers. - 4. 1 Breathers in Hamiltonian lattices. - 4. 2 Discrete breathers in one-dimensional chains. - 4. 3 Arguments for the existence of discrete breathers: some mathematics. - 4. 4 Discrete breathers: numerical experiments. - 4. 5 Discrete breathers in lattices with a homogeneous potential. - 4. 6 An exactly solvable Hamiltonian lattice. - 5 Quantum theory of vibrational - rotational spectra in XY2 - type molecules. - 5. 1 Quantum theory of two Morse oscillators: stretching vibration spectra in XY2 molecules. - 5. 2 Stretching vibration spectra: nonresonant modes' effect upon clusterization of levels. - 5. 3 Spectrum of stretching vibrations in XY2 molecules: numerical studies. - 6 Quantum Hamiltonians of vibrational rotational excitations in polyatomic molecules; method of contact transformations. - 6. 1 Quantum Hamiltonian of polyatomic molecules. - 6. 2 Molecular Hamiltonian in the inner basis representation 'coordinates -angles'. - 6. 3 Contact transformation. - 6. 4 Vibrational-rotational Hamiltonian for XY2 molecule in the inner basis representation 'coordinates - angles'. - 7 Semiclassical rotational spectra of rigid and semi-rigid molecules; clusterization of levels and internal symmetry. - 7. 1 Spectrum of polyatomic molecules: clusterization of rotational levels. - 7. 2 Rotational spectrum of rigid and semi-rigid molecules: semi-classical approach. - 7. 3 Symmetry - based analysis of rotational clusters. - 7. 4 Tunnelling matrix. - 7. 5 Violation of permutation symmetry and the super-hyperfine structure. - 8 Interaction of vibrations and rotations; polyads and their spectra. - 8. 1 Interaction of local vibration modes and rotation in hydrides. - 8. 2 Rotational - vibrational spectrum in the vicinity of bifurcation points. - 8. 3 Spectral properties of systems with Fermi resonances. - 9 Intramolecular relaxation of vibrations in the presence of local modes. - 9. 1 Localized long - living vibrational states in molecular crystals. - 9. 2 Population inversion of localized long - living vibrational states in a cold lattice. - 9. 3 Interaction of isolated levels with the continuous spectrum. - 10 IR spectra and intramolecular relaxation. - 10. 1 Redistribution of intramolecular energy: influence of multi-mode resonance. - 10. 2 Relation between the absorption linewidths intramolecular vibrations evolution and unimolecular decay rates. - 10. 3 Effect of anharmonicity on IR spectra of polyatomic molecules. - 10. 4 Correlation between structure and IR spectra in polyatomic molecules. - 11 Conclusion. Language: English