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We introduce the notions of shifted bisymplectic and shifted double Poissonstructures on differential graded associative algebras, and more generally onnon-commutative derived moduli functors with well-behaved cotangent complexes.For smooth algebras concentrated in degree $0$, these structures recover theclassical notions of bisymplectic and double Poisson structures, but in generalthey involve an infinite hierarchy of higher homotopical data, ensuring thatthey are invariant under quasi-isomorphism. The structures induce shiftedsymplectic and shifted Poisson structures on the underlying commutative derivedmoduli functors, and also on underlying representation functors.  We show that there are canonical equivalences between the spaces of shiftedbisymplectic structures and of non-degenerate $n$-shifted double Poissonstructures. We also give canonical shifted bisymplectic and bi-Lagrangianstructures on various derived non-commutative moduli functors of modules overCalabi--Yau dg categories.

Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks