Severa and Roytenberg observed that Courant algebroids are in one-to-one correspondence with differential graded (DG) symplectic manifolds of degree 2. I will describe this correspondence, as well as an integration procedure (due to Severa, following Sullivan) involving mapping spaces. The result of the integration procedure is a symplectic 2-groupoid, but it is infinite-dimensional. Nonetheless, in the case of exact Courant algebroids, the process can be explicitly carried out and described in ordinary terms. This construction gives a nice conceptual explanation for why (twisted) Dirac structures integrate to (twisted) presymplectic groupoids. This talk is based on joint work with Xiang Tang (arXiv:1310.6587).