In Chap. 5 , we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with discrete state spaces. Such processes are regenerative, and sequential return moments in any state are regeneration moments for them. We present necessary and sufficient conditions of finiteness and give upper bounds for power moments of return times supplemented by some solidarity assertions and necessary and sufficient conditions of finiteness and upper bounds exponential moments of return times using the method of test functions. Then, we obtain ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes with discrete state spaces. Also, we present an example of semi-Markov versions of the Hastings-Metropolis MCMC algorithm and comment on a possible interpretation of the above ergodic theorems applied to the corresponding semi-Markov process as distributional semi-Markov versions of this algorithm.