A “permanent” dynamical system is one whose positive solutions stay bounded away from zero and infinity. The permanence property has important applications in biochemistry, cell biology, and ecology. Inspired by reaction network theory, we define a class of polynomial dynamical systems called tropically endotactic. We show that these polynomial dynamical systems are permanent, irrespective to the values of (possibly time-dependent) parameters in these systems. These results generalize the permanence of 2D reversible and weakly reversible mass-action systems.