  In this paper we present an approach to quadratic structures in derived
algebraic geometry. We define derived n-shifted quadratic complexes, over
derived affine stacks and over general derived stacks, and give several
examples of those. We define the associated notion of derived Clifford algebra,
in all these contexts, and compare it with its classical version, when they
both apply. Finally, we prove three main existence results for derived shifted
quadratic forms over derived stacks, define a derived version of the
Grothendieck-Witt group of a derived stack, and compare it to the classical
one.
