Pantev, Toën, Vaquié and Vezzosi have introduced the notion of a shifted symplectic form, extending classical symplectic geometry to the very general context of derived stacks. An essential feature of these structures is that the usual notions of closure and nondegeneracy for a two-form hold only up to homotopy. As a result, shifted symplectic forms tend to be quite intricate, involving many components related by many equations. I will explain how, in certain cases, shifted symplectic forms can be strictified to obtain concrete models in terms of more classical geometric structures, such as Dirac structures and Courant algebroids twisted by gerbes. This talk is based on joint work with Pavel Safronov.