Dilute ferromagnetic oxides having Curie temperatures far in excess of 300 K and exceptionally large ordered moments per transition-metal cation challenge our understanding of magnetism in solids. These materials are high-k dielectrics with degenerate or thermally activated n-type semiconductivity. Conventional super-exchange or double-exchange interactions cannot produce long-range magnetic order at concentrations of magnetic cations of a few percent. We propose that ferromagnetic exchange here, and in dilute ferromagnetic nitrides, is mediated by shallow donor electrons that form bound magnetic polarons, which overlap to create a spin-split impurity band. The Curie temperature in the mean-field approximation varies as (xdelta)(1/2) where x and delta are the concentrations of magnetic cations and donors, respectively. High Curie temperatures arise only when empty minority-spin or majority-spin d states lie at the Fermi level in the impurity band. The magnetic phase diagram includes regions of semiconducting and metallic ferromagnetism, cluster paramagnetism, spin glass and canted antiferromagnetism.