All species experience temporal fluctuations in environmental conditions e.g. temperature, precipitation, or mortality risk. These fluctuations often are autocorrelated e.g. warmer years tending to be followed by warmer years. How these autocorrelations influence ecological dynamics remains,  largely, an open problem. As a step toward tackling this problem, I review mathematical techniques for characterizing long-term ecological outcomes for stochastic difference equations. These techniques rely on non-zero Lyapunov exponents (growth rates when rare) at stationary distributions of the models. Using diffusion style approximations of these exponents, I illustrate how autocorrelated fluctuations can  promote metapopulation persistence, can  generate stochastic priority effects for species sharing a common resource, and can mediate coexistence of species sharing a predator. I will discuss how these effects stem from covariances between demographic and environmental processes, and the relative time scales at which these processes operate.