Understanding under what conditions populations, whether they be plants, animals,
or viral particles, persist is an issue of theoretical and practical importance in
population biology. Both biotic interactions and environmental fluctuations are key factors
that can facilitate or disrupt persistence. One approach to examining the interplay between
these deterministic and stochastic forces is the construction and analysis of stochastic
difference equations $X_{t+1}=F(X_t,\xi_{t+1})$ where $X_t \in \R^k$ represents the state
of the populations and $\xi_1,\xi_2,...$ is a sequence of random variables representing
environmental stochasticity. In the analysis of these stochastic models, many theoretical
population biologists are interested in whether the models are bounded and persistent.
Here, boundedness asserts that asymptotically $X_t$ tends to remain in compact sets. In
contrast, persistence requires that $X_t$ tends to be "repelled" by some "extinction set"
$S_0\subset \R^k$. Here, results on both of these proprieties are reviewed for single
species, multiple species, and structured population models. The results are illustrated
with applications to stochastic versions of the Hassell and Ricker single species models,
Ricker, Beverton-Holt, lottery models of competition, and lottery models of
rock-paper-scissor games. A variety of conjectures and suggestions for future research are
presented.