A homologous series of ternary rare-earth zinc arsenides, prepared by reactions of the elements at 750 °C, has been identified with the formula RE(2-y)Zn4As4·n(REAs) (n = 2, 3, 4) for various RE members. They adopt trigonal structures: RE(4-y)Zn4As6 (RE = La-Nd), space group R3̄m1, Z = 3; RE(5-y)Zn4As7 (RE = Pr, Nd, Sm, Gd), space group P3̄m1, Z = 1; RE(6-y)Zn4As8 (RE = La-Nd, Sm, Gd), space group R3̄m1, Z = 3. The Zn atoms can be partially substituted by Mn atoms, resulting in quaternary derivatives RE(2-y)Mn(x)Zn(4-x)As4·n(REAs). Single-crystal structures were determined for nine ternary and quaternary arsenides RE(2-y)M4As4·n(REAs) (M = Mn, Zn) as representative examples of these series. The structures are built by stacking close-packed nets of As atoms, sometimes in very long sequences, with RE atoms occupying octahedral sites and M atoms occupying tetrahedral sites, resulting in an intergrowth of [REAs] and [M2As2] slabs. The recurring feature of all members of the homologous series is a sandwich of [M2As2]-[REAs]-[M2As2] slabs, while rocksalt-type blocks of [REAs] increase in thickness between these sandwiches with higher n. Similar to the previously known related homologous series REM(2-x)As2·n(REAs) which is deficient in M, this new series RE(2-y)M4As4·n(REAs) exhibits deficiencies in RE to reduce the electron excess that would be present in the fully stoichiometric formulas. Enthalpic and entropic factors are considered to account for the differences in site deficiencies in these two homologous series. Band structure calculations indicate that the semiconducting behaviour of the parent n = 0 member (with CaAl2Si2-type structure) gradually evolves, through a narrowing of the gap between valence and conduction bands, to semimetallic behaviour as the number of [REAs] blocks increases, to the limit of n = ∞ for rocksalt-type REAs.