Kisil, Anastasia V.; orcid: 0000-0001-7652-5880; email: anastasia.kisil@manchester.ac.ukAbrahams, I. DavidMishuris, Gennady; orcid: 0000-0003-2565-1961Rogosin, Sergei V.; orcid: 0000-0002-7356-16562021-10-202021-10-202021-10-202021-07-01Proceedings of the Royal Society A, volume 477, issue 2254, page 20210533http://hdl.handle.net/10034/626140From The Royal Society via Jisc Publications RouterHistory: received 2021-07-01, accepted 2021-09-10, pub-electronic 2021-10-20, pub-print 2021-10-27Article version: VoRPublication status: PublishedFunder: Royal Society; Id: http://dx.doi.org/10.13039/501100000288; Grant(s): Dorothy Hodgkin Research Fellowship, Wolfson Research Merit Award and Ser Cymru FutureFunder: Belarusian Republican Foundation for Fundamental Research; Id: http://dx.doi.org/10.13039/100007595; Grant(s): F20MS-083Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266; Grant(s): EP/R014604/1This paper reviews the modern state of the Wiener–Hopf factorization method and its generalizations. The main constructive results for matrix Wiener–Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener–Hopf technique.Licence for VoR version of this article: http://creativecommons.org/licenses/by/4.0/Special featureReview articlesWiener–HopfRiemann–Hilbertfactorizationpartial indicesRiemann boundary value problemapplicationsarticle2021-10-20