We introduce a new class of continuous distributions called the generalized transmuted-G family which extends the quadratic rank trans- mutation map pioneered by Shaw and Buckley (2007). We provide six special models of the new family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The fiexibility of the proposed family is illustrated by means of three applications to real data sets.