The chapter is an introduction to the book devoted to studying ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes and multi-alternating regenerative processes with semi-Markov modulation. These ergodic theorems are obtained using coupling ergodic theorems for regenerative processes presented in the first volume, the method of artificial regeneration, and the method of test functions. The introduction informally presents the main problems, methods, and algorithms that comprise the book’s contents. We give simple examples, illustrated by 12 figures, and try to show the logic and ideas underlying the method of test functions and the method of artificial regeneration and their applications for obtaining ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes and multi-alternating processes with semi-Markov modulation, as well as to explain the meaning of the results presented in the book.