We study the focusing nonlinear Schrodinger equation with finite energy and finite variance initial data. While considering the mass-supercritical regime we investigate solutions above the energy (or mass-energy) threshold, i.e., when the nergy of the solution exceeds the energy of the so-called ground state. We extend the known scattering versus blow-up dichotomy above the energy threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, characterizing invariant sets of solutions (with either scattering or blow-up in finite time behavior) possibly with arbitrary large mass and energy. We investigate other dispersive equations in a similar manner.