The Topp-Leone (TL) distribution is introduced by Topp and Leone [1]. Its probability density function is a simple function with only one parameter. Even though the TL distribution has been discussed and applied in many research fields, but there is a limitation about its shape. In this article, we propose the T-TL family of distributions using quantile function of  family of distributions to generate generalized TL distributions including the Weibull-TL{exponential}, the log-logistic-TL{exponential}, the logistic-TL{extreme value}, the exponential-TL{log-logistic} and the normal-TL{logistic} distributions. Some associated properties and inferences are discussed. Some graphical representations related to the probability density function are shown. Finally, 3 real datasets are applied to illustrate the generalized TL distributions. HIGHLIGHTS The Topp-Leone distribution is introduced in 1955. Its probability density function is a simple function with only one parameter. It has been discussed and applied in many research fields The T-Topp-Leone family of distributions using quantile function of family of distributions to generate generalized Topp-Leone distributions including the Weibull-Topp-Leone{exponential}, the log-logistic-Topp-Leone{exponential}, the logistic-Topp-Leone{extreme value}, the exponential-Topp-Leone{log-logistic} and the normal-Topp-Leone{logistic} distributions Some statistical properties, such as reliability function, hazard function, quantile function of T-Topp-Leone family, Shannon entropy, moments, mean deviation and median deviation are discussed All generalized Topp-Leone distributions are applied to three real datasets and the results indicated that five distributions obtained from the new family can be used as good alternatives to the Topp-Leone, beta and Kumaraswamy distributions GRAPHICAL ABSTRACT