We investigate the role of structure and chemical composition on the uptake of poly(ethylene oxide) by a series of graphite oxides (GOs) and thermally reduced GOs, leading to the formation of polymer-intercalated GO and polymer-adsorbed graphene nanostructures. To this end, a series of poly(ethylene oxide) (PEO) - GO hybrid materials exhibiting a variable degree of GO oxidation and exfoliation has been investigated in detail using a combination of techniques including X-ray photoelectron spectroscopy, X-ray diffraction, thermogravimetry, scanning-electron microscopy, and nitrogen adsorption. Intercalation of the polymer phase into well-defined GO galleries is found to correlate well with both the degree of GO oxidation and with the presence of hydroxyl groups. The latter feature is an essential prerequisite to optimize polymer uptake owing to the predominance of hydrogen-bonding interactions between intercalant and host. Unlike the bulk polymer, these intercalation compounds show neither crystallisation nor glass-transition associated with the polymer phase. Exfoliation and reduction of GO result in high-surface-area graphene layers exhibiting the highest polymer uptake in these GO-based materials. In this case, PEO undergoes surface adsorption, where we observe the recovery of glass and melting transitions associated with the polymer phase albeit at significantly lower temperatures than the bulk