Sudden, persistent changes in ecosystem state or configuration, known in ecology as regime shifts, are difficult to predict and a cause of great concern. A large, stable prey population may suddenly collapse to an alternative low-density state in response to a stochastic perturbation, for example, or stochasticity may trigger outbreaks in pest populations that were previously stably suppressed. To explain phenomena like these, ecologists have drawn heavily on theory that emphasizes the deterministic feedbacks that destabilize equilibrium states, while marginalizing the complex role of stochasticity in driving transitions between states.  In this talk, I will discuss how different types of tipping points arise, and approaches for deriving stronger stability concepts that allow us to move beyond classical deterministic theory and draw on our understanding of transient dynamics.  Given the pervasive influence of large perturbations in nature, this view promises to yield improved insights into the factors that stabilize or destabilize ecological systems.