A theme running through much of the functionalist literature in linguistics is that grammatical structure, to a considerable degree, has an 'iconic' motivation. This theme can be distilled into three rather distinct claims: (1) iconic principles govern speakers' choices of structurally available options in discourse; (2) structural options that reflect discourse-iconic principles become grammaticalized; (3) grammatical structure is an iconic reflection of conceptual structure. After presenting numerous examples from the functionalist literature in support of the idea that iconicity is widespread in language, I argue that claim (1) is irrelevant to generative grammar; claim (2), if correct, poses no challenge to generative grammar, despite a widespread belief to the contrary; and claim (3) has literally been built into standard versions of generative grammar. I go on to discuss the implications of iconic relations in language for the autonomy hypothesis and, at a more speculative level, for the evolution of language.