The aim of this paper is to describe in some detail the topography of the space occupied by functional and cognitive models. We describe the salient characteristics of functionalist, cognitivist and/or constructionist models against the general background of usage-based models, and also include in our discussion the parallel architecture framework of Jackendoff (1997, 2002a, 2002b), recently further refined as the Simpler Syntax hypothesis (Culicover & Jackendoff, 2005). A list of 36 features for the comparison of models is drawn up, and each of 11 approaches is discussed in the light of these features. Our conclusion is that although the study provides some evidence for the view that functionalist and cognitive and/or constructionist models occupy two partially distinct areas of functional/cognitive space, there is nevertheless a large group of features which are shared across all the types of model we have examined. Other groups of features allow us to distinguish between two major groups of models, one largely functionalist in its orientation, the other cognitivist and/or constructionist, the remaining models showing affinities with one or other of the two main groups, according to the feature concerned.