A knowledge of structure is crucial to the understanding of inorganic solids and polymers. Neutron and X-ray powder diffraction are two powerful complementary techniques which can be used in the structural characterisation of a variety of crystalline materials. Chemical and electrochemical oxygen intercalation techniques involving both aqueous and non-aqueous systems, have been investigated for a number of crystalline inorganic oxides. The pyrochlore structure has been discovered to be a new class of host for the chemical intercalation of oxygett and the interstitial solid solution of Ce2Zr2O7+x based on this structure-type has been investigated. Intercalation in this system is found to involve an unusual mechanism of oxygen displacement. The structures of other complex metal oxides have also been elucidated using a combination of X-ray and neutron powder diffraction, including those of Li29Zr9Nb3O40 and Li29Zr9.6Ta2.4O40. The doping behaviour of magnesium into the technologically important material lithium niobate has also been studied using these techniques. Polymer electrolytes are a class of ionically conducting solid phases formed by the dissolution of salts in ion co-ordinating macromolecules. The relationship between the crystalline and amorphous phase of the polymer-salt complex PEO3.LiCF3SO3 has been examined by variable-temperature powder X-ray diffraction. This has shed new light on the relationship between the crystalline and amorphous structures of polymer electrolytes. Finally, the crystal structure of the polymer-salt complex PEO4:RbSCN has been determined.