Recent works in unsupervised learning have emphasized the need to understand a new trend in algorithmic design, which is to influence the clustering via weights on the instance points. In this paper, we handle clustering as a constrained minimization of a Bregman divergence. Theoretical results show benefits resembling those of boosting algorithms, and bring new modified weighted versions of clustering algorithms such as k-means, expectation-maximization (EM) and k-harmonic means. Experiments display the quality of the results obtained, and corroborate the advantages that subtle data reweightings may bring to clustering.