Summary: Markov chain Monte Carlo (MCMC) methods are statistical methods designed to sample from a given measure \(\pi\) by constructing a Markov chain that has \(\pi\) as invariant measure and that converges to \(\pi\). Most MCMC algorithms make use of chains that satisfy the detailed balance condition with respect to \(\pi\); such chains are therefore reversible. On the other hand, recent work [C.-R. Hwang et al., Ann. Appl. Probab. 15, No. 2, 1433–1444 (2005; Zbl 1069.60065); T. Lelièvre et al., J. Stat. Phys. 152, No. 2, 237–274 (2013; Zbl 1276.82042); L. Rey-Bellet and K. Spiliopoulos, Nonlinearity 28, No. 7, 2081–2103 (2015; Zbl 1338.60086); Electron. Commun. Probab. 20, Paper No. 15, 16 p. (2015; Zbl 1347.60015)] has stressed several advantages of using irreversible processes for sampling. Roughly speaking, irreversible diffusions converge to equilibrium faster (and lead to smaller asymptotic variance as well). In this paper we discuss some of the recent progress in the study of nonreversible MCMC methods. In particular: i) we explain some of the difficulties that arise in the analysis of nonreversible processes and we discuss some analytical methods to approach the study of continuous-time irreversible diffusions; ii) most of the rigorous results on irreversible diffusions are available for continuous-time processes; however, for computational purposes one needs to discretize such dynamics. It is well known that the resulting discretized chain will not, in general, retain all the good properties of the process that it is obtained from. In particular, if we want to preserve the invariance of the target measure, the chain might no longer be reversible. Therefore iii) we conclude by presenting an MCMC algorithm, the SOL-HMC algorithm [the author et al., Bernoulli 22, No. 1, 60–106 (2016; Zbl 1346.60119)], which results from a nonreversible discretization of a nonreversible dynamics.