  Sampling from a complex distribution $\pi$ and approximating its intractable
normalizing constant Z are challenging problems. In this paper, 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 $\rho$. The map T does not leave the target $\pi$
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 $\pi$ while NEO-MCMC combines multiple NEO-IS
estimates of the normalizing constant and an iterated sampling-importance
resampling mechanism to sample from $\pi$. For T chosen as a discrete-time
integrator of a conformal Hamiltonian system, NEO-IS achieves state-of-the art
performance on difficult benchmarks and NEO-MCMC is able to explore highly
multimodal targets. Additionally, we provide detailed theoretical results for
both methods. In particular, we show that NEO-MCMC is uniformly geometrically
ergodic and establish explicit mixing time estimates under mild conditions.
