In Chap. 8 , we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces and distributional atoms. We introduce regenerative semi-Markov processes with D-distributional atoms with regeneration moments, which are sequential moments of jumps from states in set D, and derive relations that express the distribution functions of its regeneration time and duration of the transition period via the distribution functions of the first hitting times into the set D. Also, necessary and sufficient conditions of finiteness and upper bounds for power and exponential moments of regeneration times, and the duration of the transition period are given in terms of test functions. Finally, we present variants of defining relations for stationary distributions for a semi-Markov process with general state space and distributional atom and its accompanying Markov processes, and ergodic theorems with explicit power and exponential upper bounds for convergence rates for such semi-Markov processes.