Theoretical and applied research in clustering have often followed separate paths, with only the occasional confluence of interest. In this tutorial, we provide an overview of recent results in the theory of clustering that bridge this divide and are of interest to practitioners. We describe a new approach to selecting the initial cluster centers in the k-means algorithm, which leads both to provable approximation guarantees, and practical improvements in the quality of the clustering. We continue by explaining why the algorithm works in non-Euclidean spaces, for example, for clustering under information measures like the Kullback-Leibler divergence, and present new algorithms for these metrics. Finally, we discuss recent results on the stability of clusterings and their implication for our ability to judge the quality of a clustering.