After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras. Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods. The connection between orthogonal transformations and Clifford algebras is established in a new way, by means of Lipschitz monoids. Lipschitz monoids also allow a more efficient study of hyperbolic spaces.