Abstract: We will discuss a certain ``modular variant'' of the Mordell-Lang theorem. Suppose that we are given a family of abelian varieties parametrised by an algebraic variety $V$. Given a point $p$ on $V$, we are then interested in all points $q$ on $V$ that correspond to an abelian variety that is isogenous to the abelian variety above $p$. We call the set of all such points the isogeny orbit of $p$. Loosely speaking, this might be thought of as an analog of a divisible group. We can then ask to classify all algebraic varieties $V$, that contain a Zariski-dense isogeny orbit. If everything is defined over the complex numbers, this problem is treated by the work of Orr and more recently Richard, Yafaev. I will discuss the history of the problem and talk about joint work with Griffon, in which we begin attacking the problem over fields of positive characteristic.