I will give a basic introduction into recent work with B. Williams in which we define quasimodular forms for the orthogonal group and show how the notion interacts with the theta lift. In particular, we prove that the theta lift of a quasimodular produces a quasimodular if and only if a certain weight-depth inequality holds. In the last part of the talk we discuss how this relates to the Gromov-Witten theory of the Enriques surface and its limitation for more general K3 fibrations. No previous knowledge about modular forms is required (all notation can be reviewed).