  Annealed Importance Sampling (AIS) and its Sequential Monte Carlo (SMC)
extensions are state-of-the-art methods for estimating normalizing constants of
probability distributions. We propose here a novel Monte Carlo algorithm,
Annealed Flow Transport (AFT), that builds upon AIS and SMC and combines them
with normalizing flows (NF) for improved performance. This method transports a
set of particles using not only importance sampling (IS), Markov chain Monte
Carlo (MCMC) and resampling steps - as in SMC, but also relies on NF which are
learned sequentially to push particles towards the successive annealed targets.
We provide limit theorems for the resulting Monte Carlo estimates of the
normalizing constant and expectations with respect to the target distribution.
Additionally, we show that a continuous-time scaling limit of the population
version of AFT is given by a Feynman--Kac measure which simplifies to the law
of a controlled diffusion for expressive NF. We demonstrate experimentally the
benefits and limitations of our methodology on a variety of applications.
