This is a post-peer-review, pre-copyedit version of the paper Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy by Giuseppe Rega Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Italy Please cite this work as follows: Publisher link and Copyright information: Rega, G., Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy, Nonlinear Dynamics, 99(1), 11-34, 2020, DOI: 10.1007/s11071-019-04833-w. You can download the final authenticated version of the paper from: http://dx.doi.org/10.1007/s11071-019-04833-w. © 2020 Springer Nature Switzerland AG This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by- nc-nd/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Licensees may copy, distribute, display and perform the work and make derivative works and remixes based on it only if they give the author or licensor the credits (attribution) in the manner specified by these. Licensees may copy, distribute, display, and perform the work and make derivative works and remixes based on it only for non-commercial purposes. Licensees may copy, distribute, display and perform only verbatim copies of the work, not derivative works and remixes based on it.