The problem of rapid and automated detection of distinct market regimes is a topic of great interest to financial mathematicians and practitioners alike. In this paper, we outline an unsupervised learning algorithm for clustering financial time series into a suitable number of temporal segments (market regimes). As a special case of the above, we develop a robust algorithm that automates the process of classifying market regimes. The method is robust in the sense that it does not depend on modeling assumptions of the underlying time series, as our experiments with real data sets show. This method – dubbed the Wasserstein k-means algorithm – frames such a problem as one on the space of probability measures with finite pth moment, in terms of the p-Wasserstein distance between (empirical) distributions. We compare our Wasserstein k-means approach with more traditional clustering algorithms by studying the so-called maximum mean discrepancy scores between, and within, clusters. In both cases it is shown that the Wasserstein k-means algorithm greatly outperforms all considered alternative approaches. We demonstrate the performance of all approaches both on synthetic data in a controlled environment and on real data.