Systematic measurements of the core-electron binding energies of the rare-earth metals terbium to lutetium, and their surface and bulk oxides, have been made with a VG-ESCA3 X-ray photoelectron spectrometer coupled with a PDP/8E computer. Comparison of the measured binding energies for the core levels with accepted tubulations reveals discrepancies that cast doubt on normally accepted values for the heavier elements. Multiplet splitting of the 4f, 4d, 4p, 4s and 5s levels is observed for the metals terbium to thulium, which have open 4f-shells. The 4f and 4d spectra of pure ytterbium and lutetium metals, with closed 4f-shells, exhibit spin-orbit doublets; whereas multiplet splitting is also detected for oxidized ytterbium, while for oxidized lutetium the simple 4f and 4d doublets are retained. Satellite structure is observed in the 4p spectra; while the 3d spectra of dysprosium and terbium - accessible to excitation by AlK$\alpha $ radiation - have complex structures. Spin-orbit splitting is also observed in 5p spectra for the whole of the series terbium to lutetium. The present results agree well with available high-resolution x.p.s. data, and are also in good agreement with many-electron predictions. The existence of additional structure predicted by the correlated hole theory is confirmed, and occurs at $ca$. 26 eV to the high binding-energy side of the most intense peak in the 4d-spectra for terbium to thulium (with incomplete 4f-shells). Measurements are also reported for bulk oxide samples, and the oxygen 1s binding energies are compared for the surface oxidized metal and bulk oxides.