We investigate the effect of the structure function FL in global parton analyses of deep inelastic and related hard scattering data. We perform NLO and NNLO analyses which include the reduced cross section HERA data at high y, as well as earlier direct measurements of FL. We find that the NNLO analysis gives a better description of FL at low x than that performed at NLO. Nevertheless the data show evidence of the need of further contributions to FL, which may be of higher-twist origin. We study such corrections both phenomenologically and theoretically via a renormalon approach. The higher-twist corrections extracted from a successful fit to the data are in general agreement with the theoretical expectations, but there is still room for alternative theoretical contributions, particularly at low x and Q2. The importance of future measurements of FL is emphasized.