In this article the T-X family is introduced by givening the cumulative distribution function (CDF)  R{W(F(x))}, where R is the CDF of a random variable T, F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R, F and W functions give different families of distributions. We used the quintile functions to define the W function. Some general properties of this T-X system of distributions are studied.   A new distribution of the T-X family is derived, namely, the Generalized Weibull Uniform Log Logistic Distribution (GRU {LL}) with four parameter. Different methods of estimation are used to estimate the unknown parameters in complete and censored random samples.