Diophantine geometry is a modern-day incarnation of mathematicians' perennial interest in solving algebraic equations in integers. Its fundamental idea is to study the geometric objects defined by algebraic equations in order to understand their integral solutions ("geometry determines arithmetic"). One of its major achievements, Faltings' theorem, states that a large class of algebraic equations has only finitely many primitive integral solutions, namely those associated with smooth, proper curves of genus > 1. In the last few years, work by Dimitrov, Gao, Habegger, and myself led to rather "uniform" bounds on the number of these solutions. Using specialization techniques, these results yield also purely geometric statements like the uniform Mordell-Lang conjecture over a field of characteristic 0. If time allows, I will conclude with an outlook on unlikely intersection problems in multiplicative groups.