In this paper we give necessary and sufficient conditions for identifiability of parameters in animal and sire models of Gaussian traits, for binary and ordered categorical threshold characters, for traits following a Poisson mixed model and in frailty models for surviving data. For survival data we consider Weibull and Cox log normal frailty models with time-independent co-variates. Furthermore, we consider Cox frailty models with time-independent covariates, extended with either an extra random effect, time-dependent covariates with associated fixed effects, or time-dependent covariates with associated random effects. We show that sire models are consistent with the additive genetic infinitesimal model if and only if a normally distributed error term is included and the sire variance is less than one third of the variance of the normally distributed error term.