In this book we summarize the essential results of our efforts over the years to calculate energies, wave functions, and electromagnetic transitions of atoms as functions of the magnetic field strength from laboratory fields up to neutron star magnetic fields. Motivated by the observational evidence of huge magnetic 5 fields with strengths up to 10 T in the vicinity of white dwarf stars and of up 9 to 10 T in the vicinity of neutron stars the authors, together with coworkers and candidates for doctor and diploma degrees, have investigated this , fasci- nating quantum mechanical problem more or less continuously since 1978. The extensive tables and figures in the appendices represent the most complete data set to date in this field of research. For practical use all numbers are available by "anonymous ftp" over Internet. The first direct measurement of a neutron star magnetic field by Trum- per and his group, who observed a cyclotron feature at about 50 ke V in the spectrum of the X-ray pulsar Hercules X-I corresponding to a field strength of 8 several 10 T, stimulated investigations of atoms within the framework of the adiabatic approximation, which is well justified for such field strengths. This method and its results are discussed in Chaps. 3, 5, and 6.