We propose a new nonparametric measure of association between an arbitrary number of random vectors. The measure is based on the empirical copula process for the multivariate marginals, corresponding to the vectors, and is robust to the within-vector dependence. It is confined to the [0,1] interval and covers the entire range of dependence from vector independence to a monotone relationship element-wise. We study the properties of the new measure under several well-known copulas and provide a nonparametric estimator of the measure, along with its asymptotic theory, under fairly general assumptions. To illustrate the applicability of the new measure, we use it in applications to financial contagion, systemic risk, and portfolio choice. Specifically, we test for contagion effects between equity markets in North and South America, Europe and Asia, surrounding the financial crisis of 2008 and find strong evidence of previously unknown contagion patterns. In the context of sovereign bonds and credit default swaps, we study the evolution of systemic risk in European financial markets and uncover large differences from previous estimates. Finally, based on a real-time portfolio utilizing the new systemic risk estimates, we illustrate the implications of the new measure for portfolio choice and risk management.