This chapter addresses the computation and the experimental identification of the nonlinear dynamics of mechanical structures through the use of nonlinear modes. They enable to reduce the dynamics, for efficient computation as well as for qualitative understanding. An overview of the complex vibratory behaviours often observed when non-linearities are at hand is proposed through selected didactic examples, as well as a classification of the sources of non-linearities encountered in mechanical engineering. Then, a focus on geometrically nonlinear structures is proposed, since they show most of the classical nonlinear dynamical phenomena and are still a computation challenge. An overview of the main models of beams, plates and shells is proposed, as well as some finite-element discretisations. The case of a piezoelectric coupling with an electronic circuit is also considered. The several definitions of nonlinear modes are considered and compared, with a focus on the normal form method to compute a reduced order model. All the particular features of nonlinear modes are addressed, in the case of a simple dynamics involving a single mode, but also when several modes are strongly interacting through internal resonances. Numerical and experimental continuation methods are also addressed, as a mean of efficiently compute and identify the dynamics of the system. Finally, a few representative examples are proposed, with focuses on internal resonances, quality of reduced order models and piezoelectric structures computations.