We propose a new family of univariate distributions generated from the Weibull random variable, called a new Weibull-X family of distributions. Two special sub-models of the proposed family are presented and the shapes of density and hazard functions are investigated. General expressions for some statistical properties are discussed. For the new family, three useful characterizations based on truncated moments are presented. Three different methods to estimate the model parameters are discussed. Monti Carlo simulation study is conducted to evaluate the performances of these estimators. Finally, the importance of the new family is illustrated empirically via two real life applications.