Supervised clustering is the problem of training clustering methods to produce desirable clusterings. Given sets of items and complete clusterings over these sets, a supervised clustering algorithm learns how to cluster future sets of items in a similar fashion, typically by changing the underlying similarity measure between item pairs. This work presents a general approach for training clustering methods such as correlation clustering and k-means/spectral clustering able to optimize to task-specific performance criteria using structural SVMs. We empirically and theoretically analyze our supervised clustering approach on a variety of datasets and clustering methods. This analysis also leads to general insights about structural SVMs beyond supervised clustering. Specifically, since clustering is a NP-hard task and the corresponding training problem likewise must make use of approximate inference during training of the parameters, we present a detailed theoretical and empirical analysis of the general use of approximations in structural SVM training.