Buckling is investigated of a long thin cylindrical shell under longitudinal compression as modelled by the von Kármán–Donnell equations. Evidence is reviewed for the buckling being localized to some portion of the axial length. In accordance with this observed behaviour the equations are first approximated circumferentially by a Galerkin procedure, whereupon cross–symmetric homoclinic solutions of the resulting system of ordinary differential equations are sought in the axial direction. Results are compared with experimental and other numerical data. Excellent agreement with experiments is achieved with fewer approximating modes than other methods.