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The idea of summarizing the information contained in a large number of variables by a small number of “factors” or “principal components” has been broadly adopted in statistics. This article introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent (instead of merely uncorrelated) factors; (ii) the use of optimal transport theory and Brenier maps to obtain a robust and efficient computational algorithm; (iii) the use of a new multivariate additive entropy decomposition to determine the most informative principal nonlinear components, and (iv) formally nesting PCA as a special case for linear Gaussian factor models. We illustrate the method’s effectiveness in an application to excess bond returns prediction from a large number of macro factors. Supplementary materials for this article are available online.