In this chapter, we present upper bounds for power and exponential moments of hitting times for semi-Markov processes obtained using the method of test functions. Hitting times and their moments play key roles in getting upper bounds for moments of coupling times and the corresponding explicit upper bounds for convergence rates in ergodic theorems for regenerative and semi-Markov-type processes. We introduce generalized hitting times and describe power transformations of such functionals, derive integral equations for power and exponential moments of Markov generalized hitting times and obtain serial representations for these moments as minimal solutions of these equations. Upper bounds for power moments of Markov generalized hitting times are given in terms of recursive test functions. Also, necessary and sufficient conditions for the finiteness of power moments for partially monotonic functionals of Markov generalized hitting times are given. Analogous results are obtained for fractional power-type and exponential moments of Markov generalized hitting times.