The processes which set fundamental limits to the minimum line widths observable in photo-electron spectroscopy are outlined. An over-all limit of about 0.002 eV is deduced as the smallest attainable line width in ordinary circumstances. A photoelectron spectrometer is described which uses focusing deflexion in a radial electrostatic field for energy analysis. This instrument has energy resolution of 0.015 eV, smaller than the normal width of rotational envelopes in spectra of polyatomic molecules. The applications to the study of molecular hydrogen and oxygen is described.