[tansey]

Stan (http://mc-stan.org/documentation/) is arguably the most advanced language. It's especially pushing the bounds of doing automatic variational inference, for the scenario where your model does not have a nice conjugate form that would be amenable to Gibbs sampling. It's not quite reached what I would say is production-quality, but some of the best people in the world of computational Bayesian methods (e.g., Michael Betancourt, most of David Blei's lab, etc.) are working on it.

[tbenst]

Yes Stan is awesome! The main difference between something like Stan and "next gen" languages like Anglican and Webppl, is there are basically no restrictions in where you use a distribution. Nested inference, probabilistic recursion, etc are all possible. For certain classes of problems this leads to greatly enhanced expressiveness. On the flip side, Stan is more production ready right now

[jmde]

A major downside of Stan is its lack of support for discrete priors. This isn't really advertised very well, but is more of a problem than it might sound initially. Its type handling also can get a little frustrating at times. Overall, I highly recommend it but it does have its downsides, and there's some room for alternatives or improvement.