If, in the mid 1980s, one had asked the average statistician about
the difficulties of using Bayesian Statistics,
his/her most likely answer would have been ''Well, there is this
problem of selecting a prior distribution and then, even if one agrees
on the prior, the whole Bayesian inference is simply impossible to
implement in practice!'' The same question asked in the 21st century
does not produce the same reply, but rather a much less serious
complaint about the lack of generic software (besides winBUGS)!
The last 
years have indeed seen a tremendous change in the way
Bayesian Statistics are perceived, both by mathematical statisticians
and by applied statisticians and the impetus behind this change has
been a prodigious leap-forward in the computational abilities. The
availability of very powerful approximation methods has correlatively
freed Bayesian modelling, in terms of both model scope and prior
modelling. As discussed below, a most successful illustration of this
gained freedom can be seen in Bayesian model choice, which was only
emerging at the beginning of the MCMC era, for lack of appropriate computational tools.
In this chapter, we will first present the most standard computational challenges met in Bayesian Statistics (Sect. 11.2), and then relate these problems with computational solutions. Of course, this chapter is only a terse introduction to the problems and solutions related to Bayesian computations. For more complete references, see Robert and Casella ([36],[37]) and [27], among others. We also restrain from providing an introduction to Bayesian Statistics per se and for comprehensive coverage, address the reader to [35], (again) among others.