The magnetism of oxides containing 5d transition metal ions differs from 3d transition metal oxides in several fundamental ways: (1) the 5d orbitals extend further from the nucleus which can lead to unexpectedly strong superexchange interactions, (2) for similar electron configurations the spin-orbit coupling of 5d ions is much stronger, and (3) competing magnetic ground states in 5d transition metal oxides are very sensitive to distortions of the lattice. In this talk I discuss our studies of the magnetic properties of A₂MM′O₆ double perovskites where M′ is either osmium, rhenium, or iridium. I will begin with double perovskites where the 5d ion is the only magnetic ion in the structure. In the latter half of the talk, I will discuss double perovskites that contain both 3d and 5d ions. A common theme throughout is an extreme sensitivity to changes in the filling of d-orbitals and relatively subtle distortions of the lattice.