In this paper the interest is to elaborate on the generalization of bivariate association measures, namely Spearman’s rho, Kendall’s tau, Blomqvist’s beta and Gini’s gamma, for a general dimension d≥2. Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii) a conical combination of all components; (iii) a set of arbitrary random components. Existing generalizations are evaluated with respect to the axiom set. For a d-variate Gini’s gamma, a simplified formula is developed, making its analytical computation easier. Further, for Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension d increases is studied. Nonparametric estimation of the considered generalizations of multivariate association measures is reviewed and a nonparametric estimator of the multivariate Gini’s gamma is introduced. The practical use of multivariate association measures is illustrated on a real data example.