This volume contains almost all mathematical papers published between 1943 and 1984 of Igor R. Shafarevich. They appear in English translations (with two exceptions, which are in French and German), some of the papers have been translated into English especially for this edition. Notes by Shafarevich at the end of the volume contain corrections and remarks on the subsequent development of the subjects considered in the papers. Igor R. Shafarevich has made a big impact on mathematics. He has worked in the fields of algebra, algebraic number theory, algebraic geometry and arithmetic algebraic geometry. His papers reflect his broad interests and include topics such as the proof of the general reciprocity law, the realization of groups as Galois groups of number fields, class field towers, algebraic surfaces (in particular K3 surfaces), elliptic curves, and finiteness results on abelian varieties, algebraic curves over number fields and lie algebras.