Inference for Bayesian models often require one to simulate from some non-standard multivariate probability distributions. In the first part of the thesis, we successfully simulate exactly from certain Bayesian posteriors (the Tobit, the constrained linear regression, smoothing spline, and the Lasso) by applying rejection sampling using exponentially tilted sequential proposal distributions. This technique is typically efficient for posteriors which have the form of truncated multivariate normal/student. In this manner, we are able to simulate exactly from the posterior in hundreds of dimensions, which has until now being unattainable. Due to the curse of dimensionality, these rejection schemes are unfortunately bound to fail as the dimensions of the problems grow. In such cases, one ultimately has to resort to approximate MCMC schemes. It is known that the sampling error of a Markov chain can be a lot easier if we can identify the regeneration times for the Markov chain. In particular, the convergence rate of a geometrically ergodic Markov chain can be estimated if one can identify the underlying regeneration events. While the idea of using regeneration in the error analysis of MCMC is not new, our contribution in the second part of the thesis is to provide simpler estimates of the total variation error, and a new graphical diagnostic with strong theoretical justification. Finally, in the third part of the thesis, we consider the exponentially tilted sequential distributions in part one as proposal distributions for the MCMC samplers in part two. We introduce a novel Reject-Regenerate sampler, which combines the lessons learned about exact sampling and regenerative MCMC into a single framework. The resulting MCMC algorithm is a Markov chain with clearly demarcated regeneration events. Moreover, in the event of a regeneration, the Markov chain achieves a perfect draw with some probability.