In this paper, we introduce a new family of univariate continuous distributions called the Gamma Kumaraswamy generalized family of distributions. We study some general mathematical properties of this family, including analysis of the shapes of the probability density and hazard rate functions, quantile function, skewness, kurtosis, linear representations of the cumulative distribution and probability density functions, moments and derived quantities, stochastic ordering, reliability parameter and order statistics. Then, we give a special attention to a particular member of the family with four parameters called the Gamma Kumaraswamy exponential distribution. Among its advantages, the corresponding probability density function can have symmetrical, left-skewed, right-skewed and reversed-J shapes, and the corresponding hazard rate function can have (near) constant, increasing, decreasing, upside-down bathtub, and bathtub shapes. Then, the inference on the Gamma Kumaraswamy exponential model is performed. The method of maximum likelihood is applied to estimate the model parameters. We illustrate the interest of the model by the analyses of two practical data sets, with favorable results in comparison to other competitive models in the filed. It is hoped that the Gamma Kumaraswamy model will be attractive for the practitioner in many applied areas.