In purely generative models, one can simulate data given parameters but not necessarily evaluate the likelihood. We use Wasserstein distances between empirical distributions of observed data and empirical distributions of synthetic data drawn from such models to estimate their parameters. Previous interest in the Wasserstein distance for statistical inference has been mainly theoretical, due to computational limitations. Thanks to recent advances in numerical transport, the computation of these distances has become feasible, up to controllable approximation errors. We leverage these advances to propose point estimators and quasi-Bayesian distributions for parameter inference, first for independent data. For dependent data, we extend the approach by using delay reconstruction and residual reconstruction techniques. For large data sets, we propose an alternative distance using the Hilbert space-filling curve, which computation scales as nlogn where n is the size of the data. We provide a theoretical study of the proposed estimators, and adaptive Monte Carlo algorithms to approximate them. The approach is illustrated on four examples: a quantile g-and-k distribution, a toggle switch model from systems biology, a Lotka-Volterra model for plankton population sizes and a L\'evy-driven stochastic volatility model.