Exploiting an overlooked observation of Blum, Kiefer & Rosenblatt (1961), Dugué (1975) and Deheuvels (1981a) described a decomposition of empirical distribution processes into a finite sum of asymptotically mutually independent terms whose limiting distribution is simple under the hypothesis that a multivariate distribution is equal to the product of its marginals. This paper revisits this idea and proposes to test independence using a combination of Cramér-von Mises statistics arising from the decomposition of the empirical copula process, which involves only the ranks of the observations. Asymptotic and finite-sample tables of critical values are provided for carrying out the test, based on Fisher's method of combining P-values. While the new statistic is inferior to the standard likelihood ratio test for multivariate normal data, Monte Carlo simulations show that it can be much more powerful than the latter when the marginal distributions of the data or their underlying dependence structure are non-normal. Using the canonical decomposition of Dugué and Deheuvels, a graphical device called a "dependogram" is also proposed which helps identify the dependence structure when the null hypothesis is rejected. The mathematical exposition, which is based on recent work of Ghoudi, Kulperger & Rémillard (2001), allows for a simultaneous treatment of the serial and non-serial case. It is shown, among other things, that the asymptotic distribution of rank statistics based on the empirical copula process is the same in both cases, thereby shedding new light on the theory of nonparametric tests of serial dependence initiated by Hallin, Ingenbleek & Puri (1985).