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The Azadkia-Chatterjee coefficient is a rank-based measure of dependence between a random variable and a random vector. In this paper, we further extend it to a measure of the dependence between two random vectors Y and Z, based on n i.i.d. samples. The proposed coefficient converges almost surely to a limit with the following properties: i) it lies in [0, 1]; ii) it is equal to zero if and only if the random vectors Y and Z are independent; and iii) it is equal to one if and only if Y is almost surely a function of Z. Remarkably, the only assumption required by this convergence is that Y is not almost surely a constant vector. We further prove that under the same mild condition and after a proper scaling, this coefficient converges in distribution to a standard normal random variable under the null hypothesis of independence. This asymptotic normality result allows us to construct a Wald-type hypothesis test of independence based on this coefficient. To compute this coefficient, we propose a merge sort based algorithm that runs in O(n (\log n)^{dim(Y)}). Finally, we show that it can be used to measure the conditional dependence between Y and Z conditional on a third random vector X, and prove that the measure is monotonic with respect to the deviation from an independence distribution under certain model restrictions.