We propose a new flexible class called the Marshall-Olkin odd Burr III family for generating continuous distributions and derive some of its statistical properties. We provide three special models which accommodate symmetrical, right-skewed and left-skewed shaped densities as well as bathtub, decreasing, increasing, reversed-J shaped and upside-down bathtub failure rate functions. The parameters are estimated by maximum likelihood, least squares and a percentile method. Some simulations investigate the accuracy of the three methods. We illustrate the utility of a special model through three applications to engineering field.