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The purpose of this paper is two fold. First to establish the theory of discounted constrained Markov decision processes with a countable state and action spaces with general multi-chain structure. Second, to introduce finite approximation methods. We define the occupation measures and obtain properties of the set of all achievable occupation measures under the different admissible policies. We establish the optimality of stationary policies for the constrained control problem, and obtain an LP with a countable number of decision variables through which stationary optimal policies are computed. Since for such an LP one cannot expect to find an optimal solution in a finite number of operations, we present two schemes for finite approximations and establish the convergence of optimal values and policies for both the discounted and the expected average cost, with unbounded cost. Sometimes it turns out to be easier to solve the problem with infinite state space than the problem with finite yet large state space. Based on the optimal policy for the problem with infinite state space, we construct policies which are almost optimal for the problem with truncated state space. This method is applied to obtain an €-optimal policy for a problem of optimal priority assignment under constraints for a system of K finite queues. [ABSTRACT FROM AUTHOR]