  We establish some deviation inequalities, moment bounds and almost sure
results for the Wasserstein distance of order p $\in$ [1, $\infty$) between the
empirical measure of independent and identically distributed R d-valued random
variables and the common distribution of the variables. We only assume the
existence of a (strong or weak) moment of order rp for some r > 1, and we
discuss the optimality of the bounds. Mathematics subject classification.
60B10, 60F10, 60F15, 60E15.
