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