Probabilistic inference algorithms for belief updating, nding the most probable explanation, the maximum a posteriori hypothesis, and the maximum expected utility are reformulated within the bucket elimination framework. This emphasizes the principles common to many of the algorithms appearing in the probabilistic inference literature and clari es the relationship of such algorithms to nonserial dynamic programming algorithms. A general method for combining conditioning and bucket elimination is also presented. For all the algorithms, bounds on complexity are given as a function of the problem's structure.