The copula function defines the degree of dependence and the structure of dependence. This paper proposes an alternative framework to decompose the dependence using quantile regression. We demonstrate that the methodology provides a detailed picture of dependence including asymmetric and non-linear relationships. In addition, changes in the degree or structure of dependence can be modeled and tested for each quantile of the distribution. The empirical part applies the framework to three different sets of financial time-series and demonstrates substantial differences in dependence patterns among asset classes and through time. The analysis of 54 global equity markets shows that detailed information about the structure of dependence is crucial to adequately assess the benefits of diversification in normal times and crisis times. © 2012 Elsevier B.V.