This book is concerned with a number of central themes in a rapidly developing area, earthquake occurrence modelling. The book starts with an introductory chapter that sketches out the state-of-the-art earthquake modelling approaches based on Markov and semi-Markov models. In Chapter 2 the long-term seismogenesis is presented in association with the evolving stress field. The occurrence times of the earthquakes are considered to have been advanced, i.e. triggered by accumulated stress changes from past nearby earthquakes and tectonic loading on the major regional faults. Chapters 3 and 4 cover the application of discrete-time hidden Markov models (HMMs) and hidden semi-Markov models (HSMMs) to earthquake occurrence data from Greece and surrounding lands. A nonparametric estimation method is used by means of which, insights into features of the earthquake process are provided which are hard to detect otherwise. Important indicators concerning the levels of the stress field are estimated by means of the suggested HMM and HSMM. Chapter 5 discusses the evaluation of the discrete-time intensity of the hitting time (DTIHT) with regard to semi-Markov chains (SMCs) and hidden Markov renewal chains (HMRCs). In addition to providing results on the evaluation of the DTIHT, it also contains formulas for DTIHT statistical estimation for both SMCs and HMRCs. This Chapter also discusses the asymptotic properties of the estimators, including strong consistency and asymptotic normality. It also contains detailed examples illustrating the theoretical results. Chapter 6 focuses on the comparison between HMMs and HSMMs in a Markov and a semi-Markov framework in order to highlight possible differences in their stochastic behavior partially governed by their transition probability matrices. Results are given in the general case where specific distributions are assumed for sojourn times as well as in the special case concerning the models applied in the previous chapters, where the sojourn time distributions are estimated non-parametrically. The impact of the differences is observed through the calculation of the mean value and the variance of the number of steps that the Markov chain (HMM case) and the EMC (HSMM case) need to make for visiting for the first time a particular state. Finally, Chapter 7 summarizes some of the most important results, provides concluding remarks and perspectives.