What can we understand about probability spaces on "nice" partitions of a geometric region? Can we design efficient samplers? Can we at least detect extreme outliers? These questions have become particularly salient in the past several years as the techniques developed by mathematicians are now applied to conduct statistical analyses of things like U.S. political districtings. We will discuss some recent developments on probability spaces defined by geometric constraints, including positive and negative results on the mixing times of relevant Markov chains, Markov chain methods which eschew mixing-time requirements, and direct sampling methods.