By a quadratic module over a commutative ring R, we mean a pair (M, q) with M a finitely generated projective R-module and q: M-t R a quadratic form. We say that (M, q) is a quadratic space if the polar form b(x, -v) = q(x + 4’) q(x) q(y) is nonsingular, i.e., induces an isomorphism M%Hom,(M, R). In [KOS] and [KP] a classification of quadratic spaces of rank 4 with trivial discriminant or more generally trivial Arf invariant is given in terms of separable quaternion algebras. In this paper we develop a similar theory for spaces of rank 6. Let A be an Azumaya algebra with an involution of symplectic type and let A, be the R-module of even elements. As remarked independently by Jacobson, Frohlich, and Tamagawa, the restriction of the reduced norm nA to A + has a square root n + . This generalizes the pfallian as a square root of the determinant for alternating matrices. If Rank,A = 16, then n, is a quadratic form on A + and (A +, n + ) is a quadratic space of rank 6. The Clifford algebra of (A + , n + ) is isomorphic to M2(A ) as a graded algebra. We recall that the Clifford algebra of a quadratic space of even rank is an Azumaya algebra, that the even Clifford algebra C, is an Azumaya algebra over its centre Z and that Z is a separable quadratic R-algebra. The isomorphism class of Z is called the Arf invariant of (M, q) and (M, q) is said to have trivial Arf invariant if Z IY R x R. Thus (A + , n + ) is a quadratic space of rank 6 with trivial Arf invariant. Furthermore, n + represents 1 since 1 E A + and n, (1) = 1. Conversely we show that if (M, q) is a quadratic space of rank 6 with trivial Arf invariant and which represents 1, then (M>q)=(A+, n +) for some Azumaya algebra A of rank 16 with a symplectic involution. This result yields classification theorems for spaces of rank 6 as well as generalizations of results of Albert and Jacobson on tensor products of quaternion algebras. A similar theory holds for forms of rank 5. Here we do not assume that the quadratic modules are nonsingular (this would imply f~ R) but only that they are semiregular. We say that a quadratic module (M, q) of rank