In any Bayesian analysis, the aim is to obtain posterior estimates for some parameters, or functions of parameters. In a limited number of cases, such estimates may be directly obtained e.g. in the case of conjugate priors. However, in general, this is not the case, and one has to resort to more indirect methods.
Before the advent of modern numerical techniques, and computing power, the necessary calculations were in practical terms impossible. However, because of the advances of technology, and due to the development of powerful numerical methods in a range of disciplines, infeasible problems of the past have become tractable.
The most important of these techniques in Bayesian statistics has been Markov chain Monte Carlo and in particular Gibbs sampling and Metropolis Hastings.