This monograph contains a comprehensive account of the recent work of the authors and other workers on large sample optimal inference for non-ergodic models. The non-ergodic family of models can be viewed as an extension of the usual Fisher-Rao model for asymptotics, referred to here as an ergodic family. The main feature of a non-ergodic model is that the sample Fisher information, appropriately normed, converges to a non-degenerate random variable rather than to a constant. Mixture experiments, growth models such as birth processes, branching processes, etc., and non-stationary diffusion...