The paper introduces distance correlation as a new measure of dependence between random vectors, which generalizes the concept of correlation in two fundamental ways: it is defined for vectors in arbitrary dimensions and characterizes independence only when the vectors are independent. The distance covariance and distance correlation are analogous to product-moment covariance and correlation, but they are more robust to departures from normality and can handle nonmonotone dependence structures. The authors derive the theoretical properties of distance covariance and correlation, including their asymptotic behavior and consistency under independence. They also propose an empirical test for multivariate independence based on these measures and present Monte Carlo results showing that the test has superior power against nonmonotone dependence compared to classical tests like the Wilks Lambda and Puri–Sen rank correlation statistics. The paper includes extensions to a one-parameter family of distance dependence measures and discusses affine invariance. Finally, the authors provide empirical results demonstrating the effectiveness of the distance covariance test in various scenarios, including multivariate normal and non-Gaussian distributions.The paper introduces distance correlation as a new measure of dependence between random vectors, which generalizes the concept of correlation in two fundamental ways: it is defined for vectors in arbitrary dimensions and characterizes independence only when the vectors are independent. The distance covariance and distance correlation are analogous to product-moment covariance and correlation, but they are more robust to departures from normality and can handle nonmonotone dependence structures. The authors derive the theoretical properties of distance covariance and correlation, including their asymptotic behavior and consistency under independence. They also propose an empirical test for multivariate independence based on these measures and present Monte Carlo results showing that the test has superior power against nonmonotone dependence compared to classical tests like the Wilks Lambda and Puri–Sen rank correlation statistics. The paper includes extensions to a one-parameter family of distance dependence measures and discusses affine invariance. Finally, the authors provide empirical results demonstrating the effectiveness of the distance covariance test in various scenarios, including multivariate normal and non-Gaussian distributions.