The discovery of the topological phase of matter has largely influenced solid state physics, photonics and acoustics research in recent years, offering not only deep physical insights into a new generation of materials and light-matter interactions, but also new engineering tools to tailor signal transport with electrons, light and sound, providing unique features in terms of robustness to defects and disorder. In recent years, we have explored opportunities to enable topologically non-trivial propagation in periodic lattices of resonators based on mechanical motion, spatio-temporal modulation and nonlinearities in the realm of optics and photonics, electromagnetics, acoustics and mechanics. Here we review our recent theoretical and experimental progress in inducing topological transitions in nonlinear arrays of resonators, and triggering the topological nature of their band properties. These transitions are associated with unusual propagation properties, including the insurgence of nonlinear solitons guided by moving domain walls, edge modes, and broadband non-reciprocal responses. The emergence of these novel topological states opens the possibility of designing novel electronic, electromagnetic, acoustic and mechanical devices with new functionalities and responses highly tolerant to imperfections in fabrication and disorder, as well as to unwanted parasitic effects.