This research project is at the crossroads of diophantine geometry and arithmetic algebraic geometry. The overall leitmotif is to understand how geometry and arithmetic are intertwined in certain objects, be these of a purely diophantine nature, such as algebraic varieties over number fields, or of an arithmetic-geometric nature, such as families of abelian varieties and Shimura varieties. The project features roughly four different lines of research. First, some problems related to the André-Oort conjecture are studied. Second, there is some follow-up work on my recent proof of the Bounded Height Conjecture for general semiabelian varieties. Third, I aim to prove a certain conjecture in (p-adic) anabelian geometry. Finally, the behavior of the Faltings height in certain settings will be investigated.