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We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructedshifted symplectic structures on derived mapping stacks having a Calabi-Yausource and a shifted symplectic target. Their construction gives a clearconceptual framework for the so-called AKSZ formalism.  We extend the PTVV construction to derived mapping stacks with boundaryconditions, which is required in most applications to quantum field theories(see e.g. the work of Cattaneo-Felder on the Poisson sigma model, and therecent work of Cattaneo-Mnev-Reshetikhin). We provide many examples ofLagrangian and symplectic structures that can be recovered in this way.  We finally give an application to topological field theories (TFTs). Weexpect that our approach will help to rigorously constuct a 2 dimensional TFTintroduced by Moore and Tachikawa. A subsequent paper will be devoted to theconstruction of fully extended TFTs (in the sense of Baez-Dolan and Lurie) frommapping stacks.

Lagrangian structures on mapping stacks and semi-classical TFTs