We introduce the Marshal-Olkin alpha power family of distributions to extend the alpha power transformed exponential distribution defined by Mahdavi and Kundu (2017) and several other distributions. The new family is analytically tractable and it can be used quite effectively for data analysis. Some structural properties of the proposed family are established. Members of the new family can have symmetrical, right-skewed and reversed-J shaped densities, and increasing, decreasing, upside-down bathtub and reversed-J shaped hazard rates. We adopt the maximum likelihood method to determined the estimates of the unknown parameters. We illustrate the performance of a special distribution in this family by means of two real data sets, which has some advantages compared to the Weibull, gamma and exponentiated exponential models.