This paper introduces a new family of continuous distributions called the transmuted transmuted-G family which extends the quadratic rank transmutation map pioneered by Shaw and Buckley [24]. We provide two special models of the new family which can be used effectively to model survival data since they accommodate increasing, decreasing, unimodal, bathtub-shaped and increasing-decreasing-increasing hazard functions. We also provide two new characterization theorems of the proposed family. The estimation of the model parameters is performed by the maximum likelihood method. The flexibility of the proposed family is illustrated by means of two applications to real data.