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The linear and nonlinear buckling of curved members is firstly studied to establish a theoretical foundation for the buckling of pipelines and rail tracks. The buckling load of circular arches and rings under hydrostatic pressure and funicular arches under compressive loading is estimated. A numerical strategy is developed to investigate the in-plane nonlinear buckling of arches. Shear deformations significantly affect the linear and nonlinear buckling behaviour. The constitutive and equilibrium equations for sandwich circular and funicular arches are also derived as this will be pertinent to the modelling of rail tracks. A new nonlinear solution strategy is presented to investigate the in-plane nonlinear localised lateral buckling of pipelines that does not make use of any usual assumptions about the buckled and adjoining regions. The critical overall pipeline length for straight pipelines and the critical included angle for circular pipelines are investigated using a localised buckling analysis. If the length or included angle is greater than or equal to a critical value, any changes to the length or included angle or changes to the boundary conditions do not influence the nonlinear localised buckling results. Both symmetric and antisymmetric localised buckling is studied for straight pipelines. For circular pipelines, the pre-buckling expansion and the post-buckling localised deformations are investigated. The present results are validated by the solutions in the literature and the ANSYS package. A sandwich model is established to investigate the in-plane nonlinear localised lateral buckling of rail tracks including the influence of sleepers and fasteners. The critical track length is investigated for the symmetric and antisymmetric buckling modes. The critical length in the antisymmetric mode is much larger than that in the symmetric mode for the same track configuration, but the safe temperature increments are almost the same for both buckling modes. The use of the fasteners with large rotational stiffness substantially increases the lateral stability of tracks. The safe temperature increment increases linearly with increasing rotational stiffness. An empirical formula is developed for the relationship between the safe temperature increment and the rotational stiffness of the fasteners. The factors that influence the localised lateral buckling of rail tracks are identified in a parametric study.