In this paper, a new family of distributions with one extra shape parameter, called the xgamma-G, is proposed. comprehensive treatment of some of its mathematical properties including ordinary and incomplete moments and quantile and generating functions are derived. The unknown model parameters are estimated by the maximum likelihood method and the performance of the maximum likelihood estimators are assessed via two extensive simulation studies. Additionally, the log-location-scale regression model for censored data based on a special member of the family is introduced. The usefulness of the proposed models is illustrated utilizing three real data sets.