MCMC (Markov Chain Monte Carlo) is the main tool in much modern computation in statistics and statistical physics. The auto-correlation time of an MCMC method is, roughly speaking, the number of MCMC "sweeps" needed to product an effectively independent sample. Variables with large correlations, for example, give long auto-correlation times in MCMC methods that move variables one at a time (single variable Metropolis), or methods that propose isotropic (i.e. uncorrelated) multivariate moves. These situations can be improved by affine transformations of the sample space, chosen to reduce correlations. We discuss two classes of MCMC samplers that are affine invariant, which means that they have the same with or without an affine pre-conditioning. One (with Jonathan Weare) involves ensembles of samples being updated together. The samplers can be affine invariant if the moves are based on relative positions of different samples. One of these is the basis of the Emcee Hammer software package distributed by Dan Foreman Mackey. The other (with Bo Zhu) is an MCMC version of Gauss Newton Marquardt Levenberg nonlinear least squares, which gets its affine invariance by using derivative information.