Because the spectral d. of Rydberg states scales with the third power of the principal quantum no. n, the region immediately below at. and mol. ionization thresholds is spectrally very dense. Studying the properties of high at. and mol. Rydberg states just below threshold (at n>100) thus represents a real exptl. challenge. At the same time, it offers the opportunity to obtain, by Rydberg series extrapolation, very detailed information on the energy level structure of mol. ions. Recent exptl. progress will be reviewed that have enabled us to obtain information on high at. and mol. Rydberg states at an unprecedented resoln. The methods we have developed rely on the use of a home-built broadly tunable near-Fourier-transform-limited pulsed extreme UV laser system with a bandwidth of better than 210 MHz and of double-resonance techniques with which the spectra of high Rydberg states can be measured at a spectral resoln. of 60 kHz. Recent applications of these methods to the study of the threshold ionization of small mol. systems will be presented. [on SciFinder (R)]