Significant progress has been made towards the generalization of some well known lifetime models, which have been successfully applied to problems arising in several areas of research. In this paper, some properties of the new Kumaraswamy exponential-Weibull (KwEW) distribution are provided. This distribution generalizes a number of well-known special lifetime models such as the Weibull, exponential, Rayleigh, modified Rayleigh, modified exponential and exponentiated Weibull distributions, among others. The beauty and importance of the new distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in environmental studies. We derive some basic properties of the KwEW distribution including ordinary and incomplete moments, skewness, kurtosis, quantile and generating functions, mean deviations and Shannon entropy. The method of maximum likelihood and a Bayesian procedure are used for estimating the model parameters. By means of a real lifetime data set, we prove that the new distribution provides a better fit than the Kumaraswamy Weibull, Marshall-Olkin exponential-Weibull, extended Weibull, exponential-Weibull and Weibull models. The application indicates that the proposed model can give better fits than other wellknown lifetime distributions.