The paper gives a bibliographical review of the finite element methods (FEMs) applied for the analysis of pressure vessel structures/components and piping from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore the aim of this review is to give the reader an encyclopaedic view of the different possibilities that exist today for the finite element analysis in the fields of pressure vessels and piping. The bibliography at the end of the paper contains approximately 1900 references to papers, conference proceedings and theses/dissertations on the subject that were published in 1976-1996. These are classified in the following categories: linear and non-linear, static and dynamic, stress and deflection analysis; stability problems; thermal problems; fracture mechanics problems: contact problems: fluid-structure interaction problems: manufacturing of pipes and tubes; welded pipes and pressure vessel components: development of special finite elements for pressure vessels and pipes; finite element software: and other topics. Also finite element software, general purpose and special purpose codes, used for the analysis of pressure vessels and pipes are briefly discussed and presented. Copyright (C) 1996 Elsevier Science Ltd.