The paper reviews the results obtained so far in the endeavour to recalculate the whole of atomic physics for magnetic field strengths so strong that the Lorentz forces are comparable with or larger than the Coulomb forces, i.e. B ⩾ 4.7.105 T·Z2 (Z = nuclear charge). Investigations of this kind are stimulated by the existence of such huge fields in the vicinity of neutron stars. Detailed results are presented for one-electron systems (including energy levels, electromagnetic bound-bound, bound-free, and freefree transitions), for two-electron systems, and for the H+2-ion