In the last 10 years, there have been major advances in the average-case analysis of bin packing, scheduling and similar partitioning problems in one and two dimensions. These problems are drawn from important applications throughout industry, often under the name of stock cutting. This article briefly surveys many of the basic results, as well as the probabilistic methods used to obtain them. The impact of the research discussed here has been twofold. First, analysis has shown that heuristic solutions often perform extremely well on average and hence can be recommended in practice, even though worst-case behavior can be quite poor. Second, the techniques of applied probability that have developed for the analysis of bin packing have found application in completely different arenas, such as statistics and stochastic models.