We talk about "unlikely intersections" whenever we have a non-empty intersection between algebraic varieties that, for dimensional reasons, we do not expect to intersect. This expectation lies behind several landmark results and conjectures in Diophantine geometry, including Faltings’ Theorem (formerly the Mordell Conjecture), the Manin-Mumford Conjecture, the André-Oort Conjecture (proved by Pila, Shankar and Tsimerman), and the still open Zilber-Pink Conjecture. In this talk, I will give an introduction to unlikely intersections, focusing first on algebraic tori and abelian varieties, and then on families of abelian varieties. I will survey some key results by Masser-Zannier and Barroero-Capuano in this setting, and finally present results from my PhD thesis establishing some partial progress on the Zilber-Pink Conjecture for curves in abelian schemes.