Markov chain Monte Carlo algorithms are used to simulate from complex statistical distributions by way of a local exploration of these distributions. This local feature avoids heavy requests on understanding the nature of the target, but it also potentially induces a lengthy exploration of this target, with a requirement on the number of simulations that grows with the dimension of the problem and with the complexity of the data behind it. Several techniques are available towards accelerating the convergence of these Monte Carlo algorithms, either at the exploration level (as in tempering, Hamiltonian Monte Carlo and partly deterministic methods) or at the exploitation level (with Rao-Blackwellisation and scalable methods).