We define and study a new generalization of the complementary Weibull geometric distribution introduced by Tojeiro et al. (J. Stat. Comput. Simul. 84 (2014) 1345–1362). The new lifetime model is referred to as the Kumaraswamy complementary Weibull geometric distribution and includes twenty three special models. Its hazard rate function can be constant, increasing, decreasing, bathtub and unimodal shaped. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, Rényi entropy, mean residual life and mean inactivity time are derived. The method of maximum likelihood is used for estimating the model parameters. We provide some simulation results to assess the performance of the proposed model. Two applications to real data sets show the flexibility of the new model compared with some nested and non-nested models.