The goal of this survey is to provide a road-map for exploring the growing area of stochastic mixed-integer programming (SMIP) models and algorithms. We provide a comprehensive overview of existing decomposition algorithms for two-stage SMIPs, including Dantzig-Wolfe decomposition, dual decomposition, Lagrangian cuts, and decomposition approaches using parametric cutting planes and scaled cuts. Moreover, we explicitly discuss the relationship between these methods. Furthermore, building on these two-stage results, we summarize recent developments in the emerging field of multistage stochastic mixed-integer programming, and we present directions for future research.