Earthquake occurrence modeling is a rapidly developing research area. This book deals with its critical issues, ranging from theoretical advances to practical applications. The introductory chapter outlines state-of-the-art earthquake modeling approaches based on stochastic models. Chapter 2 presents seismogenesis in association with the evolving stress field. Chapters 3 to 5 present earthquake occurrence modeling by means of hidden (semi- )Markov models and discuss associated characteristic measures and relative estimation aspects. Further comparisons, the most important results and our concluding remarks are provided in Chapters 6 and 7. This book is mainly intended for statisticians, mathematicians working in applied probability, and theoretically-oriented geophysicists and engineers interested in semi-Markov process theories, but it could be equally supportive to Master's and PhD students and professional researchers interested in stochastic modeling for probing the nature of earthquake phenomena.