Abstract : A synthesis of the research works of the author from the past ten years is proposed in the present document. It is divided in two main parts. The first one gathers all models and computational strategies developped by the author for solving problems of nonlinear dynamics of elastic and piezoelectric structures. The main steps for solving a mechanical problem are followed. The first step is the choice of an adapted model. Thus, we propose a synthesis of geometrically nonlinear models of slender structures, classified and compared through both their formulation and the associated assumptions. Analytical as well as numerical models are addressed and compared, from their theoretical continuum mechanics foundations to the practical use of their governing equations. Then, adapted solving methods are described: analytical model discretization by modal expansion, reduced order finite element modelling, nonlinear modes, numerical following path methods. Experimental techniques are also described. The second part gives an overview of the main results obtained in the three applications followed by the author: the nonlinear dynamics of plates and shells applied to musical instruments, the vibration reduction by means of piezoelectric shunts and the nonlinear vibrations of micro / nano electromechanical systems.