The theory of linear associative algebras probably reached its zenith when the solution was found for the problem of determining all rational division algebras. Since that time it has been my hope that I might develop a reasonably self-contained exposition of that solution as well as of the theory of algebras upon which it depends and which contains the major portion of my own discoveries. The first step in carrying out this desire was necessarily that of writing a text with contents selected so as to provide a foundation adequate for the prospective exposition. This text has already been published under the title Modern Higher Algebra. Its completion was followed shortly by the timely invitation of the Colloquium Committee of the American Mathematical Society to me to write these Lectures embodying the desired exposition.