An optically pumped far-infrared laser and superconducting magnet have been used to perform high resolution studies of the energy levels of neutral and negatively charged shallow donors in high purity n-GaAs and n-InP in magnetic fields where the dimensionless magnetic field gamma is approximately one (where gamma=hoc/(2R*), hoc is the cyclotron energy and R* is the Coulomb binding energy). The central cell structure caused by the presence of different shallow donor species has been studied on the 1s-2p+1,0 transitions of neutral shallow donors in undoped GaAs samples grown by molecular beam epitaxy, liquid phase epitaxy and vapour phase epitaxy (VPE). VPE material showed two new shallow donor species with negative central cell shifts. The ls-2p-1 transition at magnetic fields where gamma > 1 shows exceptionally well resolved central cell structure. Detailed structure at magnetic fields below the 1s-2p+2 transition is due to transitions from the is to higher excited states. Samples of undoped high purity InP grown by the VPE, metal organic chemical vapour deposition and bulk growth techniques have been studied. VPE samples always show a strong component related to sulphur though some also show a strong silicon related component, and some show up to 7 components. A bulk, sample showed two strong components shallower than silicon which may have negative central cell shifts. Transitions between the excited states of neutral shallow donors in GaAs have been studied. Recent theoretical work by Makado (1982) describes the transition energies very well. Clearly resolved central cell structure is observed on inter-excited state transitions involving the 2s state. The first unambiguous observation of negatively charged shallow donors (D-states) in GaAs is reported. Simultaneous observations of transitions involving D-states, the cyclotron resonance and inter-excited state transitions of neutral donors over a wide magnetic field range, 0.03 < gamma < 3.5, highlight the differences between the transitions and the relative effects of optical excitation, temperature, magnetic field and electric field bias.