In this paper, we show that the way internal estimates are used to measure variable importance in Random Forests are also applicable to feature selection in unsupervised learning. We propose a new method called Random Cluster Ensemble (RCE for short), that estimates the out-of-bag feature importance from an ensemble of partitions. Each partition is constructed using a different bootstrap sample and a random subset of the features. We provide empirical results on nineteen benchmark data sets indicating that RCE, boosted with a recursive feature elimination scheme (RFE), can lead to significant improvement in terms of clustering accuracy, over several state-of-the-art supervised and unsupervised algorithms, with a very limited subset of features. The method shows promise to deal with very large domains. All results, datasets and algorithms are available on line.