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The primary goal of structural health monitoring is to detect damage at its onset before it reaches a critical level. The in-depth investigation in the present work addresses deep learning applied to data-driven damage detection in nonlinear dynamic systems. In particular, autoencoders (AEs) and generative adversarial networks (GANs) are implemented leveraging on 1D convolutional neural networks. The onset of damage is detected in the investigated nonlinear dynamic systems by exciting random vibrations of varying intensity, without prior knowledge of the system or the excitation and in unsupervised manner. The comprehensive numerical study is conducted on dynamic systems exhibiting different types of nonlinear behavior. An experimental application related to a magneto-elastic nonlinear system is also presented to corroborate the conclusions., Comment: 34 pages, 17 figures, 4 tables