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.

1. Composition Algebras.- 2. The Automorphism Group of an Octonion Algebra.- 3. Triality.- 4. Twisted Composition Algebras.- 5. J-algebras and Albert Algebras.- 6. Proper J-algebras and Twisted Composition Algebras.- 7. Exceptional Groups.- 8. Cohomological Invariants.