Information in the data often has far fewer degrees of freedom than the number of variables encoding the data. Dimensionality reduction attempts to reduce the number of variables used to describe the data. In this article, we shall survey some dimension reduction techniques that are robust. We consider linear dimension reduction first and describe robust principal component analysis (PCA) using three approaches. The first approach uses a singular value decomposition of a robust covariance matrix. The second approach employs robust measures of dispersion to realize PCA as a robust projection pursuit. The third approach uses a low‐rank plus sparse decomposition of the data matrix.We also survey robust approaches to nonlinear dimension reduction under a unifying framework of kernel PCA. By using a kernel trick, the robust methods available for PCA can be extended to nonlinear cases. WIREs Comput Stat 2015, 7:63–69. doi: 10.1002/wics.1331This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning Statistical and Graphical Methods of Data Analysis > Robust Methods