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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Riemann Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface''s purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann''s inequality, the theorem reached its definitive form for Riemann surfaces after work of Riemann''s short-lived student Gustav Roch in the 1850s. It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.