Fully parametric inference is where the investigator specifies the joint distribution of the data completely apart from a fixed, finite number of unknown parameters. This distribution provides the likelihood function whose study is the basis of inference both about the unknown parameters and about the adequacy of that distribution as a model for the process generating the data. The joint distribution depends upon two factors. The first is the specification of the probability law governing the passage of individuals from state to state. For a Markov or semi-Markov process this amounts to specifying the transition intensities - how they depend upon the date, upon the elapsed duration, upon both constant and time-varying regressors, possibly including unmeasured person-specific heterogeneity. The second is the sampling scheme, in particular whether we have sampled, for example, the population of entrants to a state, the population of people regardless of their state, or the population of members of a particular state. Thus we can identify four stages in fully parametric inference.