1/Normal Oscillations in Autonomous Conservative Systems.- 1.1. Harmonic solution by the Ritz method.- 1.2. Harmonic solution by the averaging and asymptotic methods.- 1.3. Comparison of methods: nonlinear normal modes and nonlinear normal coordinates.- 1.4. Examples - theoretical and computer simulation analysis.- 1.5. Harmonic plus constant term solutions - systems with quadratic nonlinearities.- 2/Normal Oscillations of Elastic Nonlinear Continuous Systems.- 2.1. Harmonic solution - `Linear Normal Mode' approach.- 2.2. Harmonic solution - `Nonlinear Normal Mode' approach.- 2.3. The generalized Ritz method for a beam with nonlinear boundary conditions.- 3/Free Oscillations with Arbitrary Initial Conditions.- 3.1. The multi-frequency almost-periodic solution by the harmonic balance method.- 3.2. The almost-periodic multi-frequency solutions by the averaging method and the asymptotic method.- 3.3. The almost-periodic oscillations in a two-degree-freedom system.- 4/Harmonic Solution in Nonautonomous Systems and Its Local Stability.- 4.1. The Ritz method and variational coupled Hill's equations.- 4.2. The first order unstable regions by the perturbation procedure based on the Floquet theory.- 4.3. The first order unstable regions by the asymptotic and averaging method.- 4.4. First order unstable regions of harmonic solution in a two- degree-of-freedom system - theoretical and computer simulation analysis.- 4.5. Harmonic plus constant term solution - systems with quadratic nonlinearity.- 5/Principal Resonances.- 5.1. The mode shape of resonant vibrations in undamped systems.- 5.2. The single nonlinear mode method in weakly damped systems.- 5.3. The combined Ritz-averaging method.- 5.4. Some remarks on nonlinear normal coordinates.- 6/Principal Resonances - Examples of Theoretical and Computer Simulation Analysis.- 6.1. Two-degree-of-freedom systems - problem of coupling of normal coordinates.- 6.2. Three-degree-of-freedom systems - the single nonlinear mode method.- 6.3. Homogeneous system with two-degrees-of-freedom.- 7/Secondary Resonances (Periodic and Almost-Periodic).- 7.1. Harmonic balance method: steady-state solution and its local stability.- 7.2. The averaging method: steady-state solution and its local stability.- 7.3. The combined harmonic balance - averaging procedure.- 7.4. Determination of types of secondary resonances associated with given nonlinear characteristics.- 7.5. Basins of attraction of the secondary resonances.- 7.6. Two-degree-of-freedom system: steady-state secondary resonances - theoretical and simulation analysis.- 7.7. Two-degree-of-freedom system: basins of attraction.- 8/Internal Resonances.- 8.1. Interaction of principal and internal resonances by the averaging method.- 8.2. The harmonic balance method and types of internal resonances associated with given nonlinear characteristics.- 8.3. Interaction of external and internal resonances in a two- degree-of-freedom system: theoretical and simulation results.- 9/Parametric Resonances.- 9.1. Parametric resonances in linear systems - survey of methods.- 9.2. Determination of the combination parametric resonance by the harmonic balance method.- 9.3. First order parametric resonances in nonlinear systems.- 9.4. Parametric resonances in a two-degree-of-freedom system - theoretical and computer simulation analysis.- References.