The 1963 Göttingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. The 1963 Goettingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. Composition Algebras 1(24) Quadratic and Bilinear Forms 1(3) Composition Algebras. The Minimum Equation 4(3) Conjugation. Inverses 7(2) Moufang Identities. Alternative Laws 9(2) Subalgebras. Doubling 11(3) Structure and Dimension of a Composition Algebra 14(2) A Composition Algebra is Determined by its Norm 16(2) Split Composition Algebras 18(2) Center and Associating Elements 20(1) Classification over Special Fields 21(2) Historical Notes 23(2) The Automorphism Group of an Octonion Algebra 25(12) Automorphisms Leaving a Quaternion Subalgebra Invariant 25(1) Connectedness and Dimension of the Automorphism Group 26(4) The Automorphism Group is of Type G2 30(3) Derivations and the Lie Algebra of the Automorphism Group 33(2) Historical Notes 35(2) Triality 37(32) Similarities. Clifford Algebras, Spin Groups and Spinor Norms 37(5) The Principle of Triality 42(3) Outer Automorphisms Defined by Triality 45(3) Automorphism Group and Rotation Group of an Octonion Algebra 48(2) Local Triality 50(8) The Spin Group of an Octonion Algebra 58(7) Fields of Definition 65(1) Historical Notes 66(3) Twisted Composition Algebras 69(48) Normal Twisted Composition Algebras 70(9) Nonnormal Twisted Composition Algebras 79(10) Twisted Composition Algebras over Split Cubic Extensions 89(3) Automorphism Groups of Twisted Octonion Algebras 92(2) Normal Twisted Octonion Algebras with Isotropic Norm 94(5) A Construction of Isotropic Normal Twisted Octonion Algebras 99(3) A Related Central Simple Associative Algebra 102(3) A Criterion for Reduced Twisted Octonion Algebras. Applications 105(3) More on Isotropic Normal Twisted Octonion Algebras 108(2) Nonnormal Twisted Octonion Algebras with Isotropic Norm 110(2) Twisted Composition Algebras with Anisotropic Norm 112(3) Historical Notes 115(2) J-algebras and Albert Algebras 117(44) J-algebras. Definition and Basic Properties 117(5) Cross Product. Idempotents 122(3) Reduced J-algebras and Their Decomposition 125(8) Classification of Reduced J-algebras 133(8) Further Properties of Reduced J-algebras 141(4) Uniqueness of the Composition Algebra 145(4) Norm Class of a Primitive Idempotent 149(3) Isomorphism Criterion. Classification over Some Fields 152(2) Isotopes. Orbits of the Invariance Group of the Determinant 154(5) Historical Notes 159(2) Proper J-algebras and Twisted Composition Algebras 161(12) Reducing Fields of J-algebras 161(2) From J-algebras to Twisted Composition Algebras 163(4) From Twisted Composition Algebras to J-algebras 167(4) Historical Notes 171(2) Exceptional Groups 173(12) The Automorphisms Fixing a Given Primitive Idempotent 173(5) The Automorphism Group of an Albert Algebra 178(2) The Invariance Group of the Determinant in an Albert Algebra 180(2) Historical Notes 182(3) Cohomological Invariants 185(16) Galois Cohomology 185(4) An Invariant of Composition Algebras 189(2) An Invariant of Twisted Octonion Algebras 191(4) An Invariant of Albert Algebras 195(4) The Freudenthal-Tits Construction 199(1) Historical Notes 200(1) References 201(4) Index 205