The Bounded Height Conjecture of Bombieri, Masser and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian variety G there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in G. After partial work of many authors, Habegger proved the conjecture completely for both tori and abelian varieties in 2009. In my talk, I will discuss how to prove the conjecture for general semiabelian varieties.