The most basic question for the study of morphology and the mental lexicon is whether or not words are _decomposed_: informally, this is the question of whether words are represented (and processed) in terms of some kind of smaller units; that is, broken down into constituent parts. Formally, what it means to represent or process a word as decomposed or not turns out to be quite complex. One of the basic lines of division in the field classifies approaches according to whether they decompose all “complex” words (“Full Decomposition”), or none (“Full Listing”), or some but not all, according to some criterion (typical of “Dual-Route” models). However, if we are correct, there are at least three senses in which an approach might be said to be decompositional or not, with the result that ongoing discussions of what appears to be a single large issue might not always be addressing the same distinction. Put slightly differently, there is no single question of decomposition. Instead, there are independent but related questions that define current research. Our goal here is to identify this finer-grained set of questions, as they are the ones that should assume a central place in the study of morphological and lexical representation.