I will discuss joint work with Marco Maculan in which we prove the Shafarevich conjecture for a large class of irregular varieties over number fields. Our proof combines the method of Lawrence-Sawin with a big monodromy theorem from previous work with Javanpeykar, Lehn and Maculan. If time permits, I will briefly sketch at the end some recent progress which uses Hodge modules to rule out exceptional groups in the monodromy theorem.