Most insect species are rare most of the time, but populations of certain taxa exhibit dramatic fluctuations in abundance across years. These fluctuations range from highly regular, cyclical dynamics to mathematical chaos. Peaks in abundance, or “population outbreaks” are notable both for the damage they can cause in natural and planted forests and for the rich body of research and theory they have inspired focused on elucidating drivers of population fluctuations across time and space. This chapter explores some of the key mechanisms that explain the population dynamics of outbreaking species, including variation in intrinsic growth rates, lagged endogenous feedbacks linked to top-down and/or bottom-up effects, nonlinearities in the density dependent relationship, and the existence of multiple stable and unstable equilibria, among others. We explore some basic mathematical and graphical approaches to modeling and representing these dynamics and provide a suite of empirical examples from the recent and historical literature.