The Fluid Dynamics Panel of AGARD conducted a Symposium on "Progress and Challenges in CFD Methods and Algorithms" in Seville, Spain, on October 2-5, 1995. The purpose of this symposium was to identify and discuss topics which are likely to constitute pacing items and challenges in Computational Fluid Dynamics. Sessions were devoted specifically to parallel computing, advanced discretization schemes and advanced grid structures. Topics also include adaptive meshes, fast iterative methods and algorithmic aspects for the computation of reacting flows and unsteady flows. In this evaluation report an attempt is made to point out the critical issues for each particular subject and to assess how far they were addressed by the confer...