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Distance covariance is a quantity to measure the dependence of two randomvectors. We show that the original concept introduced and developed bySz\'{e}kely, Rizzo and Bakirov can be embedded into a more general frameworkbased on symmetric L\'{e}vy measures and the corresponding real-valuedcontinuous negative definite functions. The L\'{e}vy measures replace theweight functions used in the original definition of distance covariance. Allessential properties of distance covariance are preserved in this newframework. From a practical point of view this allows less restrictive momentconditions on the underlying random variables and one can use other distancefunctions than Euclidean distance, e.g. Minkowski distance. Most importantly,it serves as the basic building block for distance multivariance, a quantity tomeasure and estimate dependence of multiple random vectors, which is introducedin a follow-up paper [Distance Multivariance: New dependence measures forrandom vectors (submitted). Revised version of arXiv: 1711.07775v1] to thepresent article.

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance