The Galerkin's method has been used in the literature for the approximate solution of elasticity and vibration problems. In this paper, its application to compressible fluid-flow problems is carried out. I t is found that, for compressible flow problems, Galerkin's method has two advantages over the Rayleigh-Ritz method, which was previously worked out by one of the authors. First, it is not necessary to formulate variational integrals for various new problems; therefore, these problems can be treated without any mathematical difficulty. Second, there is no restriction on the value of the ratio of specific heats, y, to be used, a fact that is necessary in the discussion of some irrotational flow problems. The nonlinear problem of compressible flow passing a circular cylinder is carried out as a numerical example. The results are found to be in good agreement with other methods. An extension of Galerkin's method to the general case of flow passing arbitrary bodies is briefly indicated. For these irrotational flow problems, however, Galerkin's method seems to suffer the disadvantage of requiring more work in carrying it out than the Rayleigh-Ritz method. The application of the Galerkin's method to laminar boundary-layer problems is discussed, and the boundary-layer problem of a compressible flow past a flat plate is formulated. A numerical example is carried out to illustrate the method.