First, we give an introduction to the notion of moduli stack of a dg category. Then we explain what shifted symplectic structures are and how they are connected to Calabi-Yau structures on dg categories. More concretely, we will show that the cotangent complex to the moduli stack of a dg category A is isomorphic to the moduli stack of the *Calabi-Yau completion* of A, answering a conjecture of Keller-Yeung. This is joint work with Damien Calaque and Tristan Bozec arxiv.org/abs/2006.01069