Introduction.- Part I: Conservative systems.- Two coupled oscillators.- Two-particle systems under conditions of sonic vacuum.- Emergence and bifurcations of LPTs in the chain of three coupled oscillators.- Quasi-one-dimensional nonlinear lattices.- Part II: Extensions to non-conservative systems.- Duffing oscillators.- Non-conventional synchronization of weakly coupled active oscillators.- Limiting phase trajectories and the emergence of autoresonance in anharmonic oscillators.- Part III: Applications.- Targeted energy transfer.- Nonlinear energy channeling in the 2D, locally resonant, systems.- Nonlinear Targeted Energy Transfer and Macroscopic Analogue of the Quantum Landau-Zener Effect in Coupled Granular Chains.- Forced pendulum.- Classical analog of linear and quasi-linear quantum tunneling.- Locally supported nonlinear lattices.- Nonlinear vibrations of the carbon nanotubes.- Conclusions.