In this paper, we advance new families of bivariate copulas constructed by distributional distortions of existing bivariate copulas. The distortions under consideration are based on the unit gamma distribution of two forms. When the initial copula is Archimedean, the induced copula is also Archimedean under the admissible parameter space. Properties such as Kendall’s tau coefficient, tail dependence coefficients and tail orders for the new families of copulas are derived. An empirical application to economic indicator data is presented.