This chapter discusses a commonly used method of finding maximum-likelihood estimates of hidden Markov models (HMM) is the EM algorithm. It presents two- and three-state models fitted by the EM algorithm to the earthquakes data. The EM algorithm is an iterative method for performing maximum likelihood estimation when some of the data are missing, and exploits the fact that the complete-data log-likelihood may be straightforward to maximize even if the likelihood of the observed data is not. As J. Bulla and A. Berzel point out, researchers and practitioners tend to use either EM or direct numerical maximization, but not both, to perform maximum likelihood estimation in HMMs, and each approach has its merits. However, one of the merits claimed for EM in some generality turns out to be illusory in the context of HMMs. O. Cappe et al. provide a discussion of the relative merits of EM and direct maximization of the likelihood of an HMM by gradient-based methods.