The book presents reduction methods that are using tools from dynamical systems theory in order to provide accurate models for nonlinear dynamical solutions occurring in mechanical systems featuring either smooth or non smooth nonlinearities. The cornerstone of the chapters is the use of methods defined in the framework of the invariant manifold theory for nonlinear systems, which allows definitions of efficient methods generating the most parsimonious nonlinear models having minimal dimension, and reproducing the dynamics of the full system under generic assumptions. Emphasis is put on the development of direct computational methods for finite element structures. Once the reduced order model obtained, numerical and analytical methods are detailed in order to get a complete picture of the dynamical solutions of the system in terms of stability and bifurcation. Applications from the MEMS and aerospace industry are covered and analyzed. Geometric nonlinearity, friction nonlinearity and contacts in jointed structures, detection and use of internal resonance, electromechanical and piezoelectric coupling with passive control, parametric driving are surveyed as key applications. The connection to digital twins is reviewed in a general manner, opening the door to the efficient use of invariant manifold theory for nonlinear analysis, design and control of engineering structures.