The synthesis, structure, and acid function of solid phosphoric acid (SPA) catalyst were studied in detail. P-31 and Si-29 MAS NMR and X-ray powder diffraction identified the following crystalline silicon phosphate phases in SPA : Si5O(PO4)(6), hexagonal-SiP2O7, Si(HPO4)(2). H2O, and SiHP3O10. The acidity of SPA is due to a liquid or glassy solution of phosphoric acid oligomers supported on the silicon phosphate phases. N-15 MAS NMR of adsorbed pyridine-N-15 and C-13 MAS NMR of adsorbed acetone-2-C-13 showed Bronsted acid sites and no Lewis acid sites. H-1-->N-15-->P-31 and H-1-->C-13-->P-31 double cross polarization MAS NMR of the probe molecules provided a rare opportunity to use NMR to unambiguously localize chemisorption sites; the probe molecules are complexed to phosphoric acid and pyrophosphoric acid but not to the silicon phosphate phases. In situ NMR of the oligomerization of propene on SPA suggests that propene quantitatively reacts with phosphoric acid and its oligomers to form isopropyl phosphate, and formation of this very stable intermediate accounts for the lower olefin oligomerization activity of SPA relative to acidic zeolites. Theoretical calculations including geometries at B3LYP/6-311+G(d,p) and chemical shifts at GIAO-MP2/tzp/dz were used to model complexation of acetone or propene to SPA, and these support our conclusions.