Clustering procedures are notoriously short on rigorous guarantees. In this tutorial, I will cover some of the types of analysis that have been applied to clustering, and emphasize open problems that remain. Part I. Approximation algorithms for clustering Two popular cost functions for clustering are k-center and k-means. Both are NP-hard to optimize exactly. (a) Algorithms for approximately optimizing these cost functions. (b) Hierarchical versions of such clusterings. (c) Clustering when data is arriving in a streaming or online manner. Part II. Analysis of popular heuristics (a) How good is k-means? How fast is it? (b) Probabilistic analysis of EM. (c) What approximation ratio is achieved by agglomerative heuristics for hierarchial clustering? Part III. Statistical theory in clustering What aspects of the underlying data distribution are captured by the clustering of a finite sample from that distribution? (a) Consistency of k-means. (b) The cluster tree and linkage algorithms. (c) Rates for vector quantization.