The structure, in particular symmetry, geometry and morphology of organic chelates coordinated with the lanthanide ions are analyzed in the present review. This is the first part of a complete presentation of a theoretical description of the properties of systems, which are widely used in technology, but most of all, in molecular biology and medicine. The discussion is focused on the symmetry and geometry of the cages, since these features play a dominant role in the spectroscopic activity of the lanthanides caged by organic chelates. At the same time, the spectroscopic properties require more formal presentation in the language of Racah algebra, and deserve a separate analysis. In addition to the parent systems of DOTA, DOTP, EDTMP and CDTMP presented here, their modifications by various antennas are analyzed. The conclusions that have a strong impact upon the theory of the energy transfer and the sensitized luminescence of these systems are based on the results of numerical density functional theory calculations.