Chapter 5 is devoted specifically to the history of the head-complement parameter. The first explicit proposals in this respect are found in Graffi (1980), Stowell (1981), and Travis (1984). Then, attention is focused on Kayne’s (1994) Linear Correspondence Axiom and its contribution to the crisis of the head-complement parameter. After considering Chomsky’s (1995a, 1995b) Bare Phrase Structure theory, the discussion turns to the two current main hypotheses about head directionality: on the one hand, that linearization applies in the PF component, as proposed by Richards (2004, 2008); on the other hand, that linear order is determined within narrow syntax, as put forth by Biberauer and Roberts (2015) and Roberts (2019). The chapter ends with a review of Donati and Branchini’s (2013) experimental perspective on linearization, which supports the idea that linear order is part of externalization rather than narrow syntax.