A manifold is Calabi-Yau in the algebro-geometric sense if it admits a holomorphic orientation. Fixing such an orientation as part of the structure, and using Serre duality, the morphism spaces in the derived category of coherent sheaves on the manifold are endowed with bilinear pairings satisfying special anti-symmetry and cyclicity properties. Abstract ing from this, one arrives at the notion of Calabi-Yau dg category, a non-commutative version of Calabi-Yau manifold, which arises not only in algebraic geometry, but also in homotopy theory, symplectic topology, and representation theory. We shall discuss how a Calabi-Yau structure on a finite type dg category endows the moduli space of objects in the category with a 'shifted symplectic structure', generalising results of Pantev-Toen-Vaquie-Vezzosi. This is joint work in progress with Tobias Dyckerhoff.