An approach is suggested for testing whether the dependence structure of a random sample of multivariate data is appropriately modelled by a given family of copulas. The test procedures stem from an application of the multivariate probability integral transformation. The proposed statistics are functionals of an empirical process that is shown to be weakly convergent and asymptotically normal for various copula models commonly met in practice. Simulations are used to study the empirical behavior of the proposed goodness-of-fit tests in finite samples, and their application is illustrated on two classical sets of multivariate data.