Over the last couple of decades, several copula based methods have been proposed in the literature to test for the independence among several random variables. But these existing tests are not invariant under monotone transformations of the variables, and they often perform poorly if the dependence among the variables is highly non-monotone in nature. In this article, we propose a copula based measure of dependency and use it to construct some new distribution-free tests of independence. The proposed measure and the resulting tests, all are invariant under permutations and monotone transformations of the variables. Our dependency measure involves a kernel function, and we use the Gaussian kernel for that purpose. We adopt a multi-scale approach, where we look at the results obtained for several choices of the bandwidth parameter associated with the Gaussian kernel and aggregate them judiciously. Large sample properties of the dependency measure and the resulting tests are derived under appropriate regularity conditions. Several simulated and real data sets are analyzed to compare the performance of the proposed tests with some popular tests available in the literature.