In this talk, I will present recent results in complex dynamics about holomorphic maps with integer or rational multipliers. These were motivated by a conjecture by Milnor stating that power maps, Chebyshev maps and flexible Lattes maps are the only rational maps that have an integer multiplier at each cycle. This conjecture was recently proved by Ji and Xie. I will present a variant of their proof, which also shows that power maps and Chebyshev maps are the only entire maps whose multipliers all lie in a given discrete subring of $\mathbb{C}$. I will also present a stronger version of Milnor's conjecture stating that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattes map. In contrast, there exist transcendental entire maps whose multipliers all lie in a given number field.