We report on electron transport in oxide heterostructures with superconducting/magnetic(S/M) thin film interfaces. The investigated hybrid mesa-heterostructuresconsist of a cuprate superconductor, a nonsuperconducting cuprate(antiferromagnetic) or manganite (ferromagnetic) interlayer with thicknessdM = 5–50 nm and a conventional superconductor (Nb). The superconducting critical current(IC) with a criticalcurrent density jc = 103 A cm − 2 (for dM = 10 nm) and acharacteristic voltage ICRN = 100–200 µV (RN is normal resistance) are observed at liquid helium temperature for aCaXSr1 − XCuO2 antiferromagnetic cuprate interlayer with a thickness ofdM = 10–50 nm. The superconducting current-phase relation of heterostructures deviates from theregular sine type, demonstrating a second harmonic component. These hybridheterostructures with S/M interfaces show unusually high sensitivity to external magneticfields. When substituting the cuprate interlayer by a manganite film, no critical current wasobserved although the manganite interlayer was made several times thinner (down todM ≤ 5 nm).