Recently, there have been conceptually novel developments in Monte Carlo methods through the introduction of new MCMC algorithms which are based on continuous-time, rather than discrete-time, Markov processes. These show promise for scalable Bayesian Analysis: they naturally have non-reversible dynamics which enable them to mix faster in high-dimensional settings; sometimes they can be implemented in a way that requires access to only a small number of data points at each iteration, and yet still sample from the true posterior; and they automatically take account of sparsity in the dependence structure. This talk will give an overview of the recent work in this area.