The paper introduces distance covariance and distance correlation as new measures of dependence between random vectors in arbitrary dimensions. These measures generalize the classical product-moment covariance and correlation, and they characterize independence: distance correlation is zero if and only if the random vectors are independent. The authors define a new concept of covariance with respect to a stochastic process, showing that population distance covariance coincides with covariance with respect to Brownian motion. This leads to the introduction of Brownian covariance, which is equal to distance covariance and has a simple computing formula. The advantages of using Brownian covariance and correlation over classical Pearson methods are discussed, including their ability to detect nonlinear and nonmonotone dependence. The paper also provides theoretical foundations for these measures based on characteristic functions and introduces a test statistic for independence that is consistent against all types of dependent alternatives with finite second moments. Examples illustrate the effectiveness of the distance covariance test in detecting nonlinear and nonmonotone dependencies.The paper introduces distance covariance and distance correlation as new measures of dependence between random vectors in arbitrary dimensions. These measures generalize the classical product-moment covariance and correlation, and they characterize independence: distance correlation is zero if and only if the random vectors are independent. The authors define a new concept of covariance with respect to a stochastic process, showing that population distance covariance coincides with covariance with respect to Brownian motion. This leads to the introduction of Brownian covariance, which is equal to distance covariance and has a simple computing formula. The advantages of using Brownian covariance and correlation over classical Pearson methods are discussed, including their ability to detect nonlinear and nonmonotone dependence. The paper also provides theoretical foundations for these measures based on characteristic functions and introduces a test statistic for independence that is consistent against all types of dependent alternatives with finite second moments. Examples illustrate the effectiveness of the distance covariance test in detecting nonlinear and nonmonotone dependencies.