The first detailed investigation of the crystalline R-phases in the Mg-Zn-Al ternary system is reported, which will assist with the interpretation and understanding of similar quasicrystalline phases in this phase diagram. Although ''Mg-32(Zn,Al)(49)'' was originally reported by Bergman and Pauling in 1956, some details regarding its structure and the atomic arrangement remain to be solved. Single-phase product can be obtained from reaction compositions "Mg-1.63(ZnxAl1-x)(3.37)'', 0.35 less than or equal to x less than or equal to 0.65. A combination of single-crystal X-ray diffraction, powder neutron diffraction, energy dispersive spectroscopy, densities, and theoretical modeling is needed to elucidate a complete structural model for these phases. Single-crystal X-ray diffaction gave the R-phase structure (space group Im (3) over bar, Z = 32) for three difference samples: Mg1.76(1)Zn1.46(6)Al1.65(3) (a = 14.364(3) Angstrom), Mg1.75(1)Zn1.80(2)Al1.31(1) (a = 14.212(1) Angstrom), and Mg1.73(1)Zn2.46(6)Al0.69(3) (a = 14.131(1) Angstrom). Neutron powder diffraction on three similar bulk samples gave Mg1.76(1)Zn1.34(5)Al1.76(5) (a = 14.2697(1) Angstrom), Mg1.75(1)Zn1.76(5)Al1.34(6) (a = 14.1804(1) Angstrom), and Mg1.73(1)Zn2.06(3)Al1.09(3) (a = 14.11247(6) Angstrom). For all phases in this Mg-Zn-Al system, one crystallographic site (M4) has ca. 33% vacancies and shows a mixture of Mg and Zn atoms. Mg atoms occur next to vacancies on these sites. Theoretical calculations are used to explain the observed phase width involving Zn and Al, the trend in observed site occupancies, and the occurrence of vacancies in this structure. A range of nonbonding states near the Fermi level accounts for the phase width and allows a range in valence electron concentration between 2.07 and 2.48 for the existence of this structure. Valence electron concentrations in Mg-Zn-Al, however, necessitate the occupation of some metal-metal antibonding orbitals, which creates a driving force for vacancies. Since Mg can replace atoms in the (Zn,AI) framework, the R-phases are true intermediates between Zintl phases on one hand and Hume-Rothery intermetallic phases on the other.