A subvariety of an abelian variety does not contain a Zariski dense set of points of finite order, apart from a special class of subvarieties. This is the Manin-Mumford Conjecture and was proved by Raynaud in the 1980s. Over 10 years ago, Masser and Zannier began the study of curves in a family of abelian varieties. I will explain work in progress together with Ziyang Gao in the higher dimensional setting. Our methods involve the Pila-Zannier counting strategy in an o-minimal structure, definiability results of Peterzil-Starchenko, and height inequalities.