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A ferromagnetic insulator (FI) attached to a conventional superconductor (S) changes drastically the properties of the latter. Specifically, the exchange field at the FI/S interface leads to a splitting of the superconducting density of states. If S is a superconducting film, thinner than the superconducting coherence length, the modification of the density of states occurs over the whole sample. The coexistence of the exchange splitting and superconducting correlations in S/FI structures leads to striking transport phenomena that are of interest for applications in thermoelectricity, superconducting spintronics and radiation sensors. Here we review the most recent progress in understanding the transport properties of FI/S structures by presenting a complete theoretical framework based on the quasiclassical kinetic equations. We discuss the coupling between the electronic degrees of freedom, charge, spin and energy, under …