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In this article, we introduce a new probability distribution generator called the Lambert-<i>F</i> generator. For any continuous baseline distribution <i>F</i>, with positive support, the corresponding Lambert-<i>F</i> version is generated by using the new generator. The result is a new class of distributions with one extra parameter that generalizes the baseline distribution and whose quantile function can be expressed in closed form in terms of the Lambert <i>W</i> function. The hazard rate function of a Lambert-<i>F</i> distribution corresponds to a modification of the baseline hazard rate function, greatly increasing or decreasing the baseline hazard rate for earlier times. Herein, we study the main structural properties of the new class of distributions. Special attention is given to two particular cases that can be understood as two-parameter extensions of the well-known exponential and Rayleigh distributions. We discuss parameter estimation for the proposed models considering the moments and maximum likelihood methods. Finally, two applications were developed to illustrate the usefulness of the proposed distributions in the analysis of data from different real settings.