This chapter presents general HSMM formalism and well-known particular cases. It starts with a gentle introduction to illustrate the main notions and notations for a toy example. The chapter then introduces the maximum likelihood estimation (MLE) for HSMM, the likelihood expression and evaluation, asymptotic properties of MLE and expectation-maximization (EM) algorithm for MLE computation. It presents recent results on two topics seldom addressed for HSMMs: the definition of reliability indicators and introduction of mixed effects into HSMM components, with the latter presenting a thorough survey of the literature. Some works considered real-valued hidden states or continuous time. When the underlying chain is a Markov chain (instead of a semi-Markov chain), the observation and the hidden spaces are general and the time is discrete, then the corresponding models are known as "state space models".