We will explore and attempt to link two threads: the classical theory of inequalities, with its vast literature, and the combinatorial theory of symmetric functions, which has been greatly stimulated in recent decades by its links to algebra and representation theory. From a combinatorial point of view, symmetric functions fall naturally into "families", and many classical inequalities (e.g. Newton´s inequalities, and the AGM inequality) find a natural setting within this framework. We will sketch the combinatorial foundations of this approach to symmetric function inequalities, and survey progress toward a comprehensive theory and classification.