A metal-air battery comprising: an anode comprising a metal; a cathode spaced apart from the anode; and a separator between the anode and the cathode, wherein the cathode comprises a first cathode layer comprising a first conductive material, a second cathode layer disposed on the first cathode layer, the second cathode layer comprising a second conductive material, and a gas diffusion layer disposed on a surface of the second cathode layer, wherein the first cathode layer provides a metal ion transfer path and the second cathode layer provides an electron transfer path; wherein the first conductive material and the second conductive material comprise an inorganic solid compound including a lithium-based oxide, a sodium-based oxide, or a combination thereof, wherein the first cathode layer is configured as a core structure within the second cathode layer disposed on a portion of the first cathode layer, a portion of the first cathode layer on which the second cathode layer is not disposed is in contact with the separator, and the gas diffusion layer is disposed on a surface of the second cathode layer; and wherein: the cation conductivity of the first cathode layer is greater than the electron conductivity of the first cathode layer, the electron conductivity of the second cathode layer is greater than the cation conductivity of the second cathode layer, the electron conductivity of the second cathode layer is greater than the electron conductivity of the first cathode layer, and the cation conductivity of the first cathode layer is greater than the cation conductivity of the second cathode layer. an anode comprising a metal; a cathode spaced apart from the anode; and a separator between the anode and the cathode, wherein the cathode comprises a first cathode layer comprising a first conductive material, a second cathode layer disposed on the first cathode layer, the second cathode layer comprising a second conductive material, and a gas diffusion layer disposed on a surface of the second cathode layer, a first cathode layer comprising a first conductive material, a second cathode layer disposed on the first cathode layer, the second cathode layer comprising a second conductive material, and a gas diffusion layer disposed on a surface of the second cathode layer, wherein the first cathode layer provides a metal ion transfer path and the second cathode layer provides an electron transfer path; wherein the first conductive material and the second conductive material comprise an inorganic solid compound including a lithium-based oxide, a sodium-based oxide, or a combination thereof, wherein the first cathode layer is configured as a core structure within the second cathode layer disposed on a portion of the first cathode layer, a portion of the first cathode layer on which the second cathode layer is not disposed is in contact with the separator, and the gas diffusion layer is disposed on a surface of the second cathode layer; and wherein: the cation conductivity of the first cathode layer is greater than the electron conductivity of the first cathode layer, the electron conductivity of the second cathode layer is greater than the cation conductivity of the second cathode layer, the electron conductivity of the second cathode layer is greater than the electron conductivity of the first cathode layer, and the cation conductivity of the first cathode layer is greater than the cation conductivity of the second cathode layer. the cation conductivity of the first cathode layer is greater than the electron conductivity of the first cathode layer, the electron conductivity of the second cathode layer is greater than the cation conductivity of the second cathode layer, the electron conductivity of the second cathode layer is greater than the electron conductivity of the first cathode layer, and the cation conductivity of the first cathode layer is greater than the cation conductivity of the second cathode layer. <EOS>