The focus of this talk is on finite-state hidden Markov models (HMMs) where either the transition probability matrix parameters or the observation densities (or both) change at a random changepoint. In the Bayesian setting, efficient changepoint detection strategies are proposed to minimize the average detection delay subject to false alarm constraints. The intuition developed from the theoretical study is leveraged in two application settings: detection of sudden spurts and downfalls in the activity profile of terrorist groups, and detection of changes in interaction dynamics in social media networks. Detection strategies that exploit the underlying HMM structure are shown to significantly outperform naive strategies that do not exploit this structure.