New results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field are presented. This includes a family of double well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to anti-symmetric. For generic double wells in a certain regime, tunneling is not suppressed, and a lower bound for the eigenvalue splitting is given. Joint with Fefferman and Weinstein.