Abstract Statistical models, such as Markov chains, have recently started to be studied for the purpose of Procedural Content Generation (PCG). A major problem with this approach is controlling the sampling process in order to obtain output satisfying some desired constraints. In this paper we present three approaches to constraining the content generated using multi-dimensional Markov chains: (1) a generate and test approach that simply resamples the content until the desired constraints are satisfied, (2) an approach that finds and resamples parts of the generated content that violate the constraints, and (3) an incremental method that checks for constraint violations during sampling. We test our approaches by generating maps for two classic video games, Super Mario Bros. and Kid Icarus.