This talk is about a curious phenomenon, which concerns the reliable estimation of principal components in the face of severe corruptions. Here, the scientist is given a data matrix which is the sum of an approximately low-rank matrix and a sparse matrix modeling corrupted entries. In addition, many entries may be missing. Hence, we have a blind de-mixing problem in which the goal is to recover the low-rank structure and find out which entries have been corrupted. We present a novel approach to this problem with very surprising performance guarantees as well as a few applications in computer vision and biomedical imaging, where this technique opens new perspectives.