The Levitron^TM is an axisymmetric top magnetized parallel to its symmetry axis. It can levitate above a specially constructed magnetic base if spun rapidly enough (and, as we found, not too rapidly). We have written a complete, coupled, nondissipative Hamiltonian system that describes the dynamics of the top. The system is twelfth order and is too complicated for general analysis. We have integrated the equations of motion numerically and found a region of a two-dimensional manifold of initial conditions for which levitation persists. We find that the amplitude of the dynamic variables in the region of persistent levitation scales linearly with the initial tilt of the axis with respect to gravity. We have identified three distinct modes of failure that correspond roughly to insufficient initial spin, too large an initial tilt and too great an initial spin. Our results are in general agreement with published observations and theoretical estimates.