  We present a theory of magnetic and magneto-transport phenomena at
LaAlO$_3$/SrTiO$_3$ interfaces, which as a central ingredient includes coupling
between the conduction bands and local magnetic moments originating from charge
traps at the interface. Tuning the itinerant electron density in the model
drives transitions between a heavy Fermi liquid phase with screened moments and
various magnetic states. The dependence of the magnetic phenomena on the
electron density or gate voltage stems from competing magnetic interactions
between the local moments and the different conduction bands. At low densities
only the lowest conduction band, composed of the $d_{xy}$ orbitals of Ti, is
occupied. Its antiferromagnetic interaction with the local moments leads to
screening of the moments at a Kondo scale that increases with density. However,
above a critical density, measured in experiments to be $n_c\approx 1.7\times
10^{13} cm^{-2}$, the $d_{xz}$ and $d_{yz}$ bands begin to populate. Their
ferromagnetic interaction with the local moments competes with the
antiferromagnetic interaction of the $d_{xy}$ band leading to eventual
reduction of the Kondo scale with density. We explain the distinct magneto
transport regimes seen in experiments as manifestations of the magnetic phase
diagram computed from the model. We present new data showing a relation between
the anomalous Hall effect and the resistivity in the system. The data strongly
suggests that the concentration of local magnetic moments affecting the
transport in the system is much lower than the carrier density, in accord with
the theoretical model.
