This article treats briefly the fundamental stages of the study of the spectra of atoms and ions, which constitute the fundamental problem of atomic spectroscopy. Detailed data are presented on the current state of knowledge and the level of knowledge of the spectra of atoms and ions for all elements of the periodic system. The greatest level of knowledge of the spectra has been attained in the range from H to Mo. This involves the great progress in studying high-temperature plasmas, whereby spectra of 50-fold ionization have been obtained. One can see from analyzing the level of study of the spectra that 1063 spectra had been studied by December 1981, which amounts to 19.5% of the total number for the 104 elements. The presented data on the level of study of the spectra enable an investigator to select those necessary for plasma diagnostics, and also to plan a program of new spectroscopic studies.