Summary: The proposal of more flexible distributions is an activity often required in practical contexts. In particular, adding a positive real parameter to a probability distribution by exponentiation of its cumulative distribution function has provided flexible generated distributions having interesting statistical properties. In this paper, we study general mathematical properties of a new generator of continuous distributions with three extra parameters called the exponentiated Gompertz generated (EGG) family. We present some of its special models as well as an essay on its physical motivation. From mathematical point of view, we derive explicit expressions of the EGG family: the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics, which are valid for any baseline model. We also provide a bivariate EGG extension. The estimation procedure by maximum likelihood of the new class is elaborated and discussed. In order to quantify and to assess the asymptotic behavior of this procedure, we perform a simulation study. Finally, two applications to real data are performed. Results furnish evidence in favor of the use of the EGG beta distribution as a good proposal to these data sets.