Diode effects in BaTiO3/La2CuO4, BaTiO3/SrTiO3:Nb (Nb-doped SrTiO3), (Pb,La)(Zr,Ti)O-3/Nd2CuO4, and Pb(Zr,Ti)O-3/LaNiO3 heterostructures are reported. La2CuO4 and LaNiO3 are p-type (hole conduction-type) semiconductors, and SrTiO3:Nb and Nd2CuO4 are n-type (electron conduction-type). The heterostructures were epitaxially grown to form pp, nn, and pn heterojunctions. At low applied bias, the leakage current through the heterostructures was symmetric with regard to bias polarity. With increasing applied bias, it became asymmetric and exhibited a reproducible memory effect at a forward bias. This behavior was observed for many combinations of ferroelectrics and perovskite semiconductors. A model is presented to explain the polarity of the current characteristics and its temperature dependence.