This paper concerns studies on continuous-time controlled Markov chains, that is, continuous-time Markov decision processes with a denumer-able state space, with respect to the discounted cost criterion. The cost and transition rates are allowed to be unbounded and the action set is a Borel space. We first study control problems in the class of determini stic station-ary policies and give very weak conditions under which the exi stence of ε-optimal (ε≥0) policies is proved using the construction of a minimum Q-process. Then we further consider control problems in the class of ran-domized Markov policies for (1) regular and (2) nonregular Q-processes. To study case (1), first we present a new necessary and sufficient condition for a nonhomogeneous Q-process to be regular. This regularity condition, together with the extended generator of a nonhomogeneous Markov process. is used to prove the existence of e-optimal stationary policies. Our results for case (1) are illustrated by a Schl6gl model with a controlled diffusion. F0r case (2), we obtain a similar result using Kolmogorov' s forward equation for the min. imum Q-process and we also present an example in which our assumptions are satisfied, but those used in the previous literature fail to hold.