In this paper, Topp-Leone alpha power family of distributions is presented based on the method called alpha power transformation which is suggested by Mahdavi and Kundu (2016). The Topp-Leone alpha power family of distributions is rich and flexible for modeling data. Some statistical properties of the proposed family are studied including quantile function, moments and moment generating function, mean residual life, and order statistics. Also, the results of the paper are specialized to Topp-Leone alpha power Weibull distribution. Some of its statistical properties are obtained. Moreover, the maximum likelihood estimators of the unknown parameters of the Topp-Leone alpha power Weibull distribution are obtained. A simulation study is conducted to evaluate the performance of the maximum likelihood estimators of the parameters of the Topp-Leone alpha power Weibull distribution. Finally, a real data set is applied, and the results show that the Topp-Leone alpha power Weibull distribution is more appropriate as compared to the Topp-Leone exponential, alpha power Weibull, and Weibull distributions.