This chapter describes the use of univariate state-dependent distributions other than the Poisson, and discusses how to construct hidden Markov models (HMM) for multivariate observations, distinguishing the important special case of multinomial-like observations from other multivariate observations. A notable advantage of HMMs is the ease with which the basic model can be modified or generalized, in many different ways, to provide models for a wide range of types of observations. Normal— HMMs have been used for modelling share returns series because the resulting mixture of normals can accommodate the large values of kurtosis that are a distinctive feature of such series. HMMs have been applied by a number of authors to model the returns on financial time series. In the simplest case of a two-state HMM, the states of the Markov chain can be interpreted as calm and turbulent phases of the stock market.