If X_1,...,X_n are a random sample from a density f in , then with probability one there exists a unique log-concave maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. We exhibit an iterative algorithm for computing the estimator and show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities. Applications to classification, clustering and regression problems will be discussed, as well as recent theoretical results on the performance of the estimator. The talk will be illustrated with pictures from the R package LogConcDEAD. Co-authors: Yining Chen, Madeleine Cule, Lutz Duembgen (Bern), Robert Gramacy (Cambridge), Dominic Schuhmacher (Bern) and Michael Stewart (Sydney).