In this paper, we discuss the fundamental aspects of the semi-analytical precession nutation models that were adopted by IAU Resolutions in 2000 and 2006. We show that no significant discrepancies appear between those models (Mathews et al., J Geophys Res 107:B4, ETG 3-1 3-26, 2002, Capitaine et al., Astron Astrophys 412:567 586, 2003) and other semi-analytical solutions or the INPOP06 numerical integration (Fienga et al., Astron Astrophys 477:315 327, 2008), especially for the quadratic terms. We also report on the most recent comparisons of the models with VLBI observations. We have employed different empirical models to fit the residuals, in attempting to characterize the nature of the observed curvature. The efficiencies of those empirical models are compared and their interpretations in terms of physical mechanisms are discussed. We show that a combination of linear and 18.6-year corrections is the most credible model for explaining the currently observed residuals, but that a longer span of observations is required before the true character of the effect can be determined. We note that the predictions from the ERA-2005 theory (Krasinsky, Celest Mech Dyn Astron 96:169 217, 2006) have diverged from recent VLBI results and suggest that the empirical nature of the ERA model is responsible.