The theory of quadratic forms and the intimately related theory of symmetric bilinear forms have a long and rich history, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse. (Compare [Dickson] and [Bourbaki, 24, p. 185].) Our exposition will concentrate on the relatively recent developments which begin with and are inspired by Witt's 1937 paper ''Theorie der quadratischen Formen in beliebigen Korpern.'' We will be particularly interested in the work of A. Pfister and M. Knebusch. However, some older material will be described, particularly in ChapterII. The presentation is based on lectures by Milnor at the Institute for Advanced Study, and at Haverford College under the Phillips Lecture Program, during the Fall of 1970, as well as lectures at Princeton University in 1966