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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Riemann-Hilbert correspondence is a generalization of Hilbert''s twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups. In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension 1, and there is a correspondence between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions. Such a result was proved independently by Masaki Kashiwara (1980) and Zoghman Mebkhout (1980).