This book is devoted to studies of quasi- and pseudo-stationary phenomena for nonlinearly perturbed stochastic processes and systems. The methods used are based on exponential asymptotics for the nonlinearly perturbed renewal equation. These methods enable one to get mixed ergodic and large deviation theorems and asymptotics in these theorems for nonlinearly perturbed regenerative processes and then for nonlinearly perturbed semi-Markov processes and Markov chains. Finally, these results are applied to nonlinearly perturbed queuing systems, population dynamics and epidemic models, as well as nonlinearly perturbed risk processes. The book contains an extended bibliography of works in the area.