This Ph.D. thesis studies structured Markov models (SMMs) in the context of applied probability, stochastic modelling and applied statistics. SMMs are Markov chains on countable state spaces, whose transition rate matrices have particular structures, such as bidiagonal or tridiagonal block matrices. From a modelling point of view, applicability, robustness, simplicity, and tractability of SMMs make them a very important tool for studying complex systems. This is mainly due to the fact that for evaluation of performance measures of SMMs, there are efficient numerical analysis methods, commonly called matrix analytic methods (MAM). Some popular classes of SMMs are phase-type (PH) distributions, Markovian arrival processes (MAPs) and quasi-birth-and-death (QBD) processes. This thesis describes the outcomes of three research projects dealing with SMMs and their applications in stochastic modelling and applied statistics.