Front propagation into unstable states often mediates state transitions in spatially extended systems, in biological models and across the sciences. Classical examples include the Fisher-KPP equation for population genetics, Lotka-Volterra models for competing species, and Keller-Segel models for bacterial motion in the presence of chemotaxis. A fundamental question is to predict the speed of the propagating front as well as which new state is selected in its wake. In some cases, the propagation speed in a full nonlinear PDE model agrees with that predicted by its linearization about the unstable state, in which case we say the speed is linearly determined, and the fronts are pulled. If the nonlinear speed is faster than the linear spreading speed, we say the fronts are pushed. The marginal stability conjecture asserts that front invasion speeds are determined by the spectrum of the linearization about traveling wave solutions of the PDE model. We present a formulation and proof of the marginal stability conjecture, together with complementary results which allow one to efficiently detect the transition between pushed and pulled front propagation as system parameters vary. We illustrate the utility of our theory by applying it to Lotka-Volterra systems, Keller-Segel models with repulsive chemotaxis, a model for growth of cancer stem cell driven tumors, and a FitzHugh-Nagumo model for signal propagation in nerve fibers.