Electron delocalization in atomic and molecular systems in terms of the single particle delocalization index (Fulton, R. L. J. Phys. Chem. 1993, 97, 7516) is analyzed. The single particle delocalization index is a quantitative measure of the degree of sharing of an electron between two disjoint regions in a many electron system. The overall focus of this paper is the determination of spatial regions in atoms from which electrons are greatly delocalized. The delocalization indices from spherical volumes of various radii centered on the nucleus of interest to the volumes outside those regions show remarkably well-defined shell structure. The delocalization shell structure exhibits spatial regions in atoms, determined by the intershell. minima, in which electrons are essentially localized. The delocalization shell structure of the electrons is apparent in all the atoms in the last column, of the periodic table of the elements (Ne S-1(0), Ar S-1(0), Kr S-1(0), Xe S-1(0), and Rn S-1(0)), even in heavy atoms such as gold (Au S-2(1/2)) where by traditional methods the shell structure is not clear. The delocalization shell structure of the electrons in zinc (Zn S-1(0)) is also displayed because the expected shell structure, corresponding to principal quantum number n = 4, was not firmly established by previous indicators until atomic number 32 was reached. The values of distinct maxima of the delocalization. indices, corresponding to regions within inner shells, are quite remarkable and related to the number of the electrons in each shell quantitatively. The physical meaning of these values is that the sharing of the core electrons in each shell does not extend to outer regions. The core shell structures in carbon (C P-3(0)) and silicon (Si P-3(0)) are mimicked by means of a set of the simulated natural orbitals (based on Slater's rules) used to construct the delocaliztaion index as a function of the distance from the nucleus. The separation of the contributions of definite angular momenta to the delocalization index for Ne S-1(0), Ar S-1(0), Zn S-1(0), Xe S-1(0), and Au S-2(1/2) is carried out. Interference effects between these contributions are substantially restricted to the valence region. The values of the maxima of the contributions of definite, angular momenta to the delocalization index are in agreement with the number of the s-, p-, d-, and f-electrons in each shell quantitatively. This is in agreement with well-known fact that the valence electrons only tend to be chemically active. The delocalization shell structures around the heavy atoms in the molecules LiH, H2O, CH4, SiH4, NH3, and PH3 are also displayed and compared to those around the single atoms Li S-2(1/2), O P-3(2), C P-3(0), Si P-3(0) N S-4(3/2), and P S-4(3/2). The delocalization of the core electrons remains essentially unchanged by incorporating the atoms into the molecules. The modifications of electron delocalization in the valence region are clear, and the chemically active regions in the molecules are nicely visible.