The Weibull distribution, which is frequently used for modeling survival data, is embedded in a larger family obtained by introducing an additional shape parameter. This generalized family not only contains distributions with unimodal and bathtub hazard shapes, but also allows for a broader class of monotone hazard rates. Furthermore, the distributions in this family are analytically tractable and computationally manageable. The modeling and analysis of survival data using this family is discussed and illustrated in terms of a lifetime dataset and the results of a two-arm clinical trial.