Spectroscopic investigation of high (n >>20) molecular Rydberg states below and above the first adiabatic ionization threshold will be carried out with the aims of 1) obtaining fully resolved information on the vibrational, rotational, spin-orbit and hyperfine structures of these highly excited electronic states, 2) characterizing the role of nuclear spins in molecular photoionization, 3) determining the hyperfine structure of fundamental molecular cations at kHz resolution and accuracy by Rydberg series extrapolation, 4) measuring intervals between rovibrational levels of these molecular cations at sub MHz precision, 5) gaining a complete understanding, and providing an adequate description and classification, of angular momentum coupling (including nuclear spins) in high molecular Rydberg states, 6) testing theoretical predictions of the energy level structure of Rydberg molecules by ab initio multichannel quantum defect theory (MQDT) and of the rotational, vibrational and hyperfine levels of molecular cations by ab initio quantum chemistry and QED. The spectroscopic measurements using tunable narrow-band vacuum-ultraviolet and millimeter wave radiation sources will be performed on cold samples in supersonic beams as well as on trapped samples of translationally cold Rydberg atoms and molecules. To this end, our recent approach to trap H atoms in Rydberg states electrostatically (Hogan and Merkt, Phys. Rev. Lett. 100, 043001 (2008)) will be extended to molecules, and the possibility of transfering the trapped species from electrostatic traps to magnetic and optical traps will be explored.