This paper considers constrained Markov control processes in Borel spaces, with unbounded costs. The criterion to be minimized is a long-run expected average cost, and the constraints can be imposed on similar average costs, or on average rewards, or discounted costs or rewards. We give conditions under which the constrained problem (CP) is solvable and equivalent to an equality constrained (EC) linear program. Furthermore, we show that there is no duality gap between EC and the dual program EC* and that in fact the strong duality condition holds. Finally, we introduce an explicit procedure to solve CP in some cases which is illustrated with a detailed example.