  A growing number of generative statistical models do not permit the numerical
evaluation of their likelihood functions. Approximate Bayesian computation
(ABC) has become a popular approach to overcome this issue, in which one
simulates synthetic data sets given parameters and compares summaries of these
data sets with the corresponding observed values. We propose to avoid the use
of summaries and the ensuing loss of information by instead using the
Wasserstein distance between the empirical distributions of the observed and
synthetic data. This generalizes the well-known approach of using order
statistics within ABC to arbitrary dimensions. We describe how recently
developed approximations of the Wasserstein distance allow the method to scale
to realistic data sizes, and propose a new distance based on the Hilbert
space-filling curve. We provide a theoretical study of the proposed method,
describing consistency as the threshold goes to zero while the observations are
kept fixed, and concentration properties as the number of observations grows.
Various extensions to time series data are discussed. The approach is
illustrated on various examples, including univariate and multivariate g-and-k
distributions, a toggle switch model from systems biology, a queueing model,
and a L\'evy-driven stochastic volatility model.
