<p>A general approach to use the quantile function to estimate the parameters of distributions by computing the mean absolute deviations (MAD) around the quartiles of a truncated distribution is proposed. This quantile function is available in closed form for many distributions, including those without moments and/or variance. The computation of MAD involves computing the integral of the quantile function between the first and third quartiles. The MAD has a simple interpretation as the difference in areas under the properly folded quantile function truncated at the quartiles. The method does not require the existence of any moments. For many distributions considered, the proposed method provides closed-form solutions for the parameters. It is computationally much simpler than maximum likelihood estimation and many other methods. The method is illustrated by applying it to several distributions and real datasets, including those without some moments, such as the Levy and Pareto distributions. Extensive numerical experiments indicate that the proposed method yields highly accurate results comparable to those of other, more computationally intensive methods.</p>