I will discuss a family of Markov chain Monte Carlo (MCMC) methods whose performance is unaffected by affine tranformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particulary useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions. I will also discuss ongoing work to quantify the advantages of affine invariant schemes in terms of their favorable convergence properties as well as applications.