Cubic nonlinear Schrödinger equations (NLS) comprise a class of dispersive PDE that appear in a wide range of physical settings, including nonlinear optics and Bose—Einstein condensation. These equations admit a rich set of dynamics, including scattering/decay, blowup, and soliton solutions. A fundamental problem in dispersive PDE is the classification of general solution behaviors. In this talk, we will discuss the classification of solution behaviors at or below a mass-energy threshold determined by the ground state soliton. After reviewing the state of the field for the standard cubic NLS, we will discuss how the situation changes when one considers several important variants of this equation, including the addition of external potentials as well as higher order nonlinear perturbations. This talk will discuss several joint works, including work with B. Dodson; R. Killip and M. Visan; and C. Miao and J. Zheng.