This paper presents a new approach to the maximum-flow problem, which is based on the preflow concept introduced by Karzanov. The algorithm maintains a preflow in the original network and pushes local flow excess toward the sink along estimated shortest paths in the residual graph. The algorithm is simple and intuitive, yet it runs as fast as any other known method on dense graphs, achieving an O(n³) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, the algorithm runs in O(nm log(n²/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efficient distributed and parallel implementations. A parallel implementation running in O(n² log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach–Vishkin algorithm, which also uses n processors but requires O(n²) space. The algorithm is simple and intuitive, with natural implementations in sequential and parallel models of computation. A simple sequential implementation runs in O(n³) time, and a more complicated O(nm log(n²/m))-time sequential implementation uses the dynamic tree data structure of Sleator and Tarjan. The algorithm is proven to terminate and be correct, and it is shown to run in polynomial time. The algorithm is also shown to be applicable to the minimum-cost flow problem.This paper presents a new approach to the maximum-flow problem, which is based on the preflow concept introduced by Karzanov. The algorithm maintains a preflow in the original network and pushes local flow excess toward the sink along estimated shortest paths in the residual graph. The algorithm is simple and intuitive, yet it runs as fast as any other known method on dense graphs, achieving an O(n³) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, the algorithm runs in O(nm log(n²/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efficient distributed and parallel implementations. A parallel implementation running in O(n² log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach–Vishkin algorithm, which also uses n processors but requires O(n²) space. The algorithm is simple and intuitive, with natural implementations in sequential and parallel models of computation. A simple sequential implementation runs in O(n³) time, and a more complicated O(nm log(n²/m))-time sequential implementation uses the dynamic tree data structure of Sleator and Tarjan. The algorithm is proven to terminate and be correct, and it is shown to run in polynomial time. The algorithm is also shown to be applicable to the minimum-cost flow problem.