We develop new multi-factor dynamic copula models with time-varying factor loadings and observation-driven dynamics. The new models are highly flexible, scalable to high dimensions, and ensure positivity of covariance and correlation matrices. A closed-form likelihood expression allows for straightforward parameter estimation and likelihood inference. We apply the new model to a large panel of 100 U.S. stocks over the period 2001–2014. The proposed multi-factor structure is much better than existing (single-factor) models at describing stock return dependence dynamics in high-dimensions. The new factor models also improve one-step-ahead copula density forecasts and global minimum variance portfolio performance. Finally, we investigate different mechanisms to allocate firms into groups and find that a simple industry classification outperforms alternatives based on observable risk factors, such as size, value or momentum.