What does “nothing over and above its parts” actually mean?

Some philosophers say that a whole is “nothing over and above” its parts. Most also take general extensional mereology to be treating wholes as “nothing over and above” their parts. It is not always clear, however, what exactly is meant by the phrase “nothing over and above.” Nor is it obvious why the phrase is associated with mereology, and what purpose it serves there. In the words of Peter Van Inwagen (1994, 210): “This slippery phrase has had a lot of employment in philosophy, but what it means is never explained by its employers.” This paper sheds light on the various senses in which one might hold that a whole is “nothing over and above” its parts, and on how these senses are related both to each other and to mereology.