Modern statistics is being shaped by the recent big data revolution that captured attention of mainstream media that has somewhat unfortunately overshadowed the methodological revolution of big models. Big models allow to take the full benefit of big data, understand its structure and reflect the nuances. Big models, if fitted with enough data, allow to understand complex phenomena. Big models, if used in the Bayesian setting with appropriate prior distributions, allow to make sense of few but expensive data points. Markov chain Monte Carlo and related algorithms in statistics are the engine that made the big models and big data revolutions happen by allowing efficient approximations, optimisations and summaries where the complexity of the underlying model, or the size of available data does not allow for exact closed form computations. The recent development of algorithms in statistic has brought together experts and ideas from pure and applied mathematics, computer science, statistics, computational physics, and computational biology. In this session we aim to gather some of the leading technical experts in MCMC development from research centres in Europe (including Poland, France, and the UK) to present their work, exchange ideas and foster collaborations.