Random processes determined by the characteristics of the semi-Markov process are considered in this chapter. First is a renewal process generated by return times of a given state. The systems of equations for the distribution and expectation of them have been derived. The limit theorem for the process is formulated by adopting a theorem of the renewal theory. The limiting properties of the alternating process and integral functionals of the semi-Markov process are presented in this chapter. The chapter contains illustrative examples.