The paper introduces distance covariance and distance correlation as new measures of dependence between random vectors in arbitrary dimensions. These measures generalize classical product-moment covariance and correlation, and have the property that they are zero if and only if the vectors are independent. Distance covariance is defined as the square root of a norm involving the difference between the joint characteristic function and the product of the marginal characteristic functions. Distance correlation is a standardized version of distance covariance, and it satisfies 0 ≤ R ≤ 1, with R = 0 indicating independence. Distance covariance and distance correlation are shown to be equivalent to Brownian covariance and Brownian correlation, which are defined with respect to Brownian motion. This equivalence highlights that distance covariance is a natural extension of product-moment covariance. The paper also discusses the advantages of using distance covariance and correlation over classical Pearson covariance and correlation, particularly in detecting nonlinear and nonmonotone dependence. The paper presents the theoretical foundations of distance covariance and correlation, including their properties, asymptotic behavior, and statistical consistency. It also provides an empirical formula for computing distance covariance and correlation, which is simple and efficient. The paper demonstrates that distance covariance tests are powerful for detecting all types of dependence, and that they are consistent against alternatives with finite second moments. The paper also discusses extensions of distance covariance and correlation to other types of stochastic processes, including fractional Brownian motion. It also considers affine invariance and rank-based tests, and provides examples of applications in nonlinear regression and multivariate analysis. The paper concludes that distance covariance and correlation provide a robust and flexible tool for measuring dependence in a wide range of statistical applications.The paper introduces distance covariance and distance correlation as new measures of dependence between random vectors in arbitrary dimensions. These measures generalize classical product-moment covariance and correlation, and have the property that they are zero if and only if the vectors are independent. Distance covariance is defined as the square root of a norm involving the difference between the joint characteristic function and the product of the marginal characteristic functions. Distance correlation is a standardized version of distance covariance, and it satisfies 0 ≤ R ≤ 1, with R = 0 indicating independence. Distance covariance and distance correlation are shown to be equivalent to Brownian covariance and Brownian correlation, which are defined with respect to Brownian motion. This equivalence highlights that distance covariance is a natural extension of product-moment covariance. The paper also discusses the advantages of using distance covariance and correlation over classical Pearson covariance and correlation, particularly in detecting nonlinear and nonmonotone dependence. The paper presents the theoretical foundations of distance covariance and correlation, including their properties, asymptotic behavior, and statistical consistency. It also provides an empirical formula for computing distance covariance and correlation, which is simple and efficient. The paper demonstrates that distance covariance tests are powerful for detecting all types of dependence, and that they are consistent against alternatives with finite second moments. The paper also discusses extensions of distance covariance and correlation to other types of stochastic processes, including fractional Brownian motion. It also considers affine invariance and rank-based tests, and provides examples of applications in nonlinear regression and multivariate analysis. The paper concludes that distance covariance and correlation provide a robust and flexible tool for measuring dependence in a wide range of statistical applications.