I will explain why non-abelian free groups have the same $\forall\exists$-theory, a result first proved by Sacerdote in 1973 using combinatorial group theory, then proved in a more geometrical way by Sela using the theory of groups acting on trees (developed by Rips, Sela and others). I will also discuss some generalisations of this result to larger classes of groups, such as virtually free, hyperbolic or acylindrically hyperbolic groups.