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

Quadratic forms and Clifford algebras on derived stacks