Sampling from a complex distribution π and approximating its intractable normalizing constant Z are challenging problems.  In this talk, a novel family of importance samplers (IS) and Markov chain Monte Carlo (MCMC) samplers is derived.  Given an invertible map T, these schemes combine (with weights) elements from the forward and backward orbits through points sampled from a proposal distribution ρ. The map T does not leave the target π  invariant, hence the name NEO, standing for Non-Equilibrium Orbits. NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS estimators of expectations under π while NEO-MCMC combines multiple NEO-IS estimates of the normalizing constant and an iterated sampling-importance resampling mechanism to sample from π.