This article provides the first critical review on the discovery and development of BaNdInO 4. Exploring a new structure family of ionic conductors is an important task to develop ceramic ionic conductors. Since some A 2 BO 4 compositions exhibit high oxide-ion conductivities, we investigated ABCO 4 compositions to explore new oxideion conductors with A/B/C cation-ordered structures. Here A, B and C are cations [ionic radii: r(A) ² r(B) ² r(C)]. In 2014, we discovered a new material BaNdInO 4 which belongs to a new structure family of perovskiterelated structures. This BaNdInO 4-type structure (monoclinic, P2 1 /c) consists of alternative stacking of the A rare earth oxide unit and perovskite unit with a ¹ b ¹ c ¹ tilt system. We also discovered new materials BaRInO 4 (R = Sm, Y, Ho, Er, Yb) having the BaNdInO 4-type structure, and report their lattice parameters and anisotropic chemical expansion. Electrical conductivity of BaNdInO 4 was higher than those of BaRInO 4 (R = Sm, Y, Er). Oxide-ion conduction was dominant for BaNdInO 4 in the P(O 2) region from 3.8 © 10 ¹22 to 5.5 © 10 ¹9 atm at 858°C. Oxide-ion conductivities of Ba 1.1 Nd 0.9 InO 3.95 , BaSr 0.1 Nd 0.9 InO 3.95 and BaCa 0.2 Nd 0.8 InO 3.9 were higher than that of BaNdInO 4. Structure analyses of Ba 1.1 Nd 0.9 InO 3.95 and BaSr 0.1 Nd 0.9 InO 3.95 indicated that the excess Ba and doped Sr cations were partially substituted for Nd cation and that there existed oxygen vacancies, leading to the increase of the carrier concentration and higher oxide-ion conductivity. Following the discovery of BaNdInO 4 , BaRScO 4 (R = Nd, Eu, Y, Yb) and SrYbInO 4 were reported as new ABCO 4 materials. BaYScO 4 and BaYbScO 4 have the BaNdInO 4-type structure. BaNdScO 4 and BaEuScO 4 crystallize into the space group Cmcm, which has a higher symmetry than P2 1 /c for BaNdInO 4. SrYbInO 4 is the first example of pure oxide-ion conductors with CaFe 2 O 4-type structure. Further investigations of ABCO 4 compositions and BaNdInO 4 related materials will lead to development of materials science and solid state ionics.