This chapter describes how the likelihood of any given state-space model (SSM) can be approximated arbitrarily accurately by discretizing the state space. It also describes how the hidden Markov models (HMM) forward algorithm can be used to evaluate the approximate likelihood in a computationally feasible way and illustrates the approach using the earthquake series. The model for the earthquake series is an example of a SSM with continuous-valued state process. The state architecture of HMMs is sometimes not well suited to the problem at hand. In some examples the choice of the number of states can be straightforward, possibly using model selection criteria and residual analyses, and the interpretation of the states can be intuitive. The state process of an SSM can be continuous-valued or discrete-valued; HMMs are the special case in which the state process is discrete-valued.