  We theoretically address magnetic ordering at zero and finite temperature in
both homogeneous and trapped Bose-Bose mixtures in optical lattices. By using
Bosonic Dynamical Mean-Field Theory, we obtain the phase diagram of the
homogeneous two-component Bose-Hubbard model in a three-dimensional (3D) cubic
lattice, which features competing magnetic order of XY-ferromagnetic and
anti-ferromagnetic type in addition to the Mott and superfluid states. We show
that these magnetic phases persist also in the presence of a harmonic trap.
