The maximum concurrent flow problem (MCFP) is the optimization version of the feasibility problem in multicommodity flows. The objective is to maximize the percentage of the demands which is realizable for all commodities, subject to the capacity constraints. A fully polynomial ϵ-approximate algorithm was developed by Shahrokhi and Matula to solve the MCFP when the edge capacities are the same (the MCFP with uniform capacity). In this paper, we present an ϵ-approximate algorithm for the MCFP with uniform demand (when all demands are equal). Our ϵ-approximate algorithm employs a linear size reduction from the MCFP with uniform demand to the MCFP with uniform capacity and the fully polynomial ϵ-approximate algorithm for the MCFP with uniform capacity. The computational results indicate that the algorithm is efficient. We also present an efficient combinatorial algorithm for the MCFP in planar graphs. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.