Beta Exponentiated Weibull distribution (BEWD) is an extension of the exponentiated Weibull distribution which involves two additional shape parameters. Interestingly, the additional parameters control the tails weights of the distribution and affect skewness and kurtosis of the distribution. The five-parameter BEWD is a generalized distribution in modelling lifetimes of various industrial products. Its density and hazard curves are widely heterogeneous in their shapes. Three subfamilies of the BEWD family emerge under three parameter subspaces with the property that the members of each subfamily display similar density curves. It is found that some members of the BEWD family in one of the parameter subspaces approximately behave like a normal distribution. BEWD assumes decreasing, increasing or a bathtub behaviour. Using a sample hazard curve, and so a prior understanding of restrictions on the BEWD parameters we find estimates of parameters for fitting BEWD. These estimates based on maximum likelihood are essentially more efficient than when no such knowledge about the sample hazard curve is used. Characterizations based on truncated moments and hazard rate function are obtained. Simulation study of BEWD is performed in both ways; without the knowledge of parametric conditions and using parametric constraints and compare the results. Real data applications of the proposed approach support the better fitting of BEWD than other models. A generalization of BEWD is introduced in which a transmuted parameter is added and its behaviour is studied, named as Transmuted Beta Exponentiated Weibull Distribution (TBEWD). Different mathematical properties including moments, characteristic function, skewness, kurtosis and mode are being discussed. The transmuted parameter affects the basic characteristics, shape of density function and other properties of BEWD. Characterizations of TBEWD based on truncated moments and hazard rate function are also derived. The maximum likelihood estimation (MLE) is used to estimate the model parameters. Simulation Study is performed to test the efficiency of MLEs. Various aspects of this distribution are explored in the context of its applications, which include its subfamilies displaying reasonable similarity with regard to their hazard curves. The parametric restrictions so discovered are found useful in fitting this distribution. A number of applications of TBEWD model are also given.