Sub-sea pipelines are susceptible to premature failure caused by the corrosion of the inner wall of carbon steel pipes when conveying corrosive oils and gases. One of the cheapest ways of preventing contact between the inner wall of the carbon steel pipes and the corrosive oils and gases involves the insertion of corrosion-proof steel liners inside the host pipes. The pipelines are then expanded hydraulically to the point of yielding of the inner liners so that after removal of the hydraulic pressure a mechanical bond between the inner liners and the outer pipes is formed. However, the buckling behaviour of the inner liners, constrained by the outer pipes, under thermal loading is only partially understood. Due to the confinement from the outer pipes, the inner liners can only deflect inward. In this thesis, an understanding of the one-way buckling phenomena is built-up through sequential analyses of a simplified rigid-link model, a simplified beam-link model, a continuum liner-only model, and a continuum liner-pipe model. In the process, the fundamental differences between the one-way buckling and the classic two-way buckling are examined, and the extremely important roles of imperfections in the one-way buckling are identified. Three categories of imperfections are proposed and the concept of "critical imperfection" is defined. It is concluded that if imperfections are smaller than the critical imperfections, pipeline liners will never buckle. Detailed results for a typical pipeline liner are given and verified by finite element analyses with ABAQUS. In the finite element analyses, the partial-model simulates a single lobe of the buckled shape while the full-model treats the full circumference of a pipeline liner over a length L. The effects of nine cases having different combinations of various geometric parameters are examined. Modal shape transitions and the effects of material plasticity in both the partial- and full-models are illustrated by the snapshots of the modal shapes captured during animations using a special computer programme.