Igor R. Shafarevich is the latest outstanding contemporary mathematician to be honored by Springer-Verlag with an edition of his Collected Mathematical Papers. 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 interest 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. The volume contains almost all his mathematical papers. They appear in English translations (with two exceptions where the papers are in French and German). A substantial part 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.