The aim of this talk is to explain how model theory can be fruitfully applied to the study of the free finitely generated pro-aperiodic monoid. In particular, we view elements of the free pro-aperiodic monoid as a monoid of elementary equivalence classes of models of a first-order theory. This allows us to use general facts from model theory, such as the existence of omega-saturated models, to associate labeled linear orders to elements of the monoid. One application of this theory is a new proof of the decidability of equality of aperiodic omega-terms. These results are based on joint work with B. Steinberg, first published in STACS 2017 and Israel J. Math. 2019, and on some current work in progress. I will also make a connection with more recent work of J. Almeida, A. Costa, J. C. Costa and M. Zeitoun, Publ. Mat. 2019, who associated a different linear order to elements of free pro-aperiodic monoids.