In this paper, a new transmuted general family of distributions is introduced and studied. In particular , we obtain explicit expressions for the moments, the incomplete moments, the moment generating function, entropies measures and order statistics. Further, we introduce a bivariate extension of the new family. Some sub-models of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. We discuss the different methods of estimation of the model parameters and a simulation study is investigated for one of the special models to check the asymptotic behavior of the estimates under all estimation methods. Further, the potentiality of the family is illustrated by fitting two real data sets to the mentioned sub-models.