In Chap. 7 , we present ergodic theorems with explicit upper bounds for rates of convergence for semi-Markov processes with general state spaces and atom (a state x• for which the one-state subset \(D_{x_\bullet } = \{ x_\bullet \}\) is recurrent). We introduce a regenerative semi-Markov process with an atom and get 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 corresponding atomic set, present necessary and sufficient conditions of finiteness and upper bounds given in terms of test functions for power and exponential moments of regeneration times and the duration of the transition period. We also present variants of defining relations for stationary distributions for semi-Markov processes with general state space and atom and its accompanying Markov process and ergodic theorems with explicit power and exponential upper bounds for convergence rates for such semi-Markov processes.