Fixed-domain Gromov-Witten invariants of a complex variety are virtual counts of curves whose domain has a fixed complex structure. For Fano varieties, they are genuinely enumerative at large degrees for a large class of examples, and are conjectued to be integral and non-negative in general. In this talk, I present our work on fixed-domain Gromov-Witten invariants of Fano projective bundles, which are among the first examples where the enumerativity fails. This is work in progress with Alessio Cela.