The main goal of Chapter 11 is to demonstrate how the theory developed in the previous chapters can be used in the study of various Markov models that give rise to Markov chains with asymptotically zero drift. Some of those models are popular in stochastic modelling: random walks conditioned to stay positive, state-dependent branching processes or branching processes with migration, stochastic difference equations. In contrast with the general approach discussed here, the methods available in the literature for investigation of these models are mostly model tailored. We also introduce some new models to which our approach is applicable. For example, we introduce a risk process with surplus-dependent premium rate, which converges to the critical threshold in the nett-profit condition. Furthermore, we introduce a new class of branching processes with migration and with state-dependent offspring distributions.