Approximating Bayes in the 21st Century

The 21st century has seen an enormous growth in the development and use of approximate Bayesian methods. Such methods produce computational solutions to certain `intractable' statistical problems that challenge exact methods like Markov chain Monte Carlo: for instance, models with unavailable likelihoods, high-dimensional models, and models featuring large data sets. These approximate methods are the subject of this review. The aim is to help new researchers in particular -- and more generally those interested in adopting a Bayesian approach to empirical work -- distinguish between different approximate techniques; understand the sense in which they are approximate; appreciate when and why particular methods are useful; and see the ways in which they can can be combined.