This paper presents a survey of research results obtained by the authorand his collaborators in the areas of limit theorems for Markov-type processes andrandomly stopped stochastic processes, renewal theory and ergodic theorems forperturbed stochastic processes, quasi-stationary distributions for perturbed stochas-tic systems, methods of stochastic approximation for price processes, asymptoticexpansions for nonlinearly perturbed semi-Markov processes and applications ofthe above results to queuing systems, reliability models, stochastic networks, bio-stochastic systems, perturbed risk processes, and American-type options.