Double perovskites (DP) of the general formula Ba2MReO6, where M = Mg, Zn, and Y2/3, all based on Re6+ (5d1, t2g1), were synthesized and studied using magnetization, heat capacity, muon spin relaxation, and neutron-scattering techniques. All are cubic, Fm3̅m, at ambient temperature to within the resolution of the X-ray and neutron diffraction data, although the muon data suggest the possibility of a local distortion for M = Mg. The M = Mg DP is a ferromagnet, Tc = 18 K, with a saturation moment ∼0.3 bohr magnetons at 3 K. There are two anomalies in the heat capacity: a sharp feature at 18 K and a broad maximum centered near 33 K. The total entropy loss below 45 K is 9.68 e.u., which approaches R ln 4 (11.52 e.u.) supporting a j = 3/2 ground state. The unit cell constants of Ba2MgReO6 and the isostructural, isoelectronic analogue, Ba2LiOsO6, differ by only 0.1%, yet the latter is an anti-ferromagnet. The M = Zn DP also appears to be a ferromagnet, Tc = 11 K, μsat(Re) = 0.1 μB. In this case the heat capacity shows a somewhat broad peak near 10 K and a broader maximum at ∼33 K, behavior that can be traced to a smaller particle size, ∼30 nm, for this sample. For both M = Mg and Zn, the low-temperature magnetic heat capacity follows a T3/2 behavior, consistent with a ferromagnetic spin wave. An attempt to attribute the broad 33 K heat capacity anomalies to a splitting of the j = 3/2 state by a crystal distortion is not supported by inelastic neutron scattering, which shows no transition at the expected energy of ∼7 meV nor any transition up to 100 meV. However, the results for the two ferromagnets are compared to the theory of Chen, Pereira, and Balents, and the computed heat capacity predicts the two maxima observed experimentally. The M = Y2/3 DP, with a significantly larger cell constant (3%) than the ferromagnets, shows predominantly anti-ferromagnetic correlations, and the ground state is complex with a spin frozen component Tg = 16 K from both direct current and alternating current susceptibility and μSR data but with a persistent dynamic component. The low-temperature heat capacity shows a T1 power law. The unit cell constant of B = Y2/3 is less than 1% larger than that of the ferromagnetic Os7+ (5d1) DP, Ba2NaOsO6.