How can one perform Bayesian inference on stochastic simulators with intractable likelihoods? A recent approach is to learn the posterior from adaptively proposed simulations using neural-network based conditional density estimators. However, existing methods are limited to a narrow range of proposal distributions or require importance-weighting that can limit performance in practice. Here we present automatic posterior transformation (APT), a new approach for simulation-based inference via neural posterior estimation. APT is able to modify the posterior estimate using arbitrary, dynamically updated proposals, and is compatible with powerful flow-based density estimators. We show that APT is more flexible, scalable and efficient than previous simulation-based inference techniques and can directly learn informative features from high-dimensional and time series data.