A low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder is constructed. The investigation of the wake solutions for a variety of basic modes, Hilbert spaces, and expansion modes reveals general mathematical and physical aspects which may strongly effect the success of low‐dimensional simulations. Besides the cylinder wake, detailed information about the construction of similar low‐dimensional Galerkin methods for the sphere wake, the boundary‐layer, the flow in a channel or pipe, the Taylor–Couette problem, and a variety of other flows is given.