Large-scale magnetic fields exist in cosmic structures such as galaxies and galaxy clusters. Moreover, recent gamma ray observations have suggested the existence of magnetic fields even in void regions. In this talk I will explain how such large-scale magnetic fields can be generated in the early universe. In particular, I will discuss the possibility of cosmological magnetogenesis in the post-inflationary universe, when the inflaton oscillates around its potential minimum and the universe is effectively dominated by cold matter. I will also present a new way of constraining magnetogenesis models from the Schwinger effect. Electric fields generated along with the magnetic fields induce large conductivity in the universe via the Schwinger effect, posing a major challenge for magnetic field generation, especially for the conventional inflationary magnetogenesis models.