The existence and stability of invariant distributions for stochastically monotone processes is studied. The Knaster-Tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering. Global convergence of a monotone Markov process to its unique invariant distribution is established under an easily verified assumption. Topkis' theory of supermodular functions is applied to stochastic dynamic optimization, providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone Markov process. Applications of these results to investment theory, stochastic growth, and industry equilibrium dynamics are given.