This paper presents max-flow min-cut theorems for multicommodity flow problems and their applications to approximation algorithms. The authors establish that for any n-node multicommodity flow problem with uniform demands, the max-flow is within an O(log n) factor of the min-cut. This result is existentially optimal and provides an important analogue of the 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result has significant applications in designing approximation algorithms for NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. The flow results are also applied to path routing, network reconfiguration, communication in distributed networks, scientific computing, and rapidly mixing Markov chains. The paper begins by discussing single-commodity flow problems, where the max-flow equals the min-cut. It then moves on to multicommodity flow problems, where the max-flow is defined as the maximum fraction of each commodity that can be routed without violating capacity constraints. The min-cut is defined as the minimum ratio of the capacity of the cut to the demand of the cut. The authors show that the max-flow is always upper bounded by the min-cut for any multicommodity flow problem. The paper then discusses prior work on max-flow and min-cut for multicommodity flow problems, highlighting results for specific cases such as two commodities and planar graphs. It then presents the authors' main result, showing that for uniform multicommodity flow problems, the max-flow is within a Θ(log n) factor of the min-cut. This result is generalized to product multicommodity flow problems and directed graphs. The authors also show that the result holds for flow problems with short paths and relaxed uniformity of demands. The paper then discusses applications of these results to approximation algorithms, showing how they can be used to design polynomial-time approximation algorithms for NP-hard optimization problems. The authors also describe how the results can be used to find small cuts in graphs and how these cuts can be used to design approximation algorithms for various problems. The paper concludes with a discussion of subsequent work and open questions in the field.This paper presents max-flow min-cut theorems for multicommodity flow problems and their applications to approximation algorithms. The authors establish that for any n-node multicommodity flow problem with uniform demands, the max-flow is within an O(log n) factor of the min-cut. This result is existentially optimal and provides an important analogue of the 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result has significant applications in designing approximation algorithms for NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. The flow results are also applied to path routing, network reconfiguration, communication in distributed networks, scientific computing, and rapidly mixing Markov chains. The paper begins by discussing single-commodity flow problems, where the max-flow equals the min-cut. It then moves on to multicommodity flow problems, where the max-flow is defined as the maximum fraction of each commodity that can be routed without violating capacity constraints. The min-cut is defined as the minimum ratio of the capacity of the cut to the demand of the cut. The authors show that the max-flow is always upper bounded by the min-cut for any multicommodity flow problem. The paper then discusses prior work on max-flow and min-cut for multicommodity flow problems, highlighting results for specific cases such as two commodities and planar graphs. It then presents the authors' main result, showing that for uniform multicommodity flow problems, the max-flow is within a Θ(log n) factor of the min-cut. This result is generalized to product multicommodity flow problems and directed graphs. The authors also show that the result holds for flow problems with short paths and relaxed uniformity of demands. The paper then discusses applications of these results to approximation algorithms, showing how they can be used to design polynomial-time approximation algorithms for NP-hard optimization problems. The authors also describe how the results can be used to find small cuts in graphs and how these cuts can be used to design approximation algorithms for various problems. The paper concludes with a discussion of subsequent work and open questions in the field.