Let $(\varphi_t)$ be a semigroup of holomorphic self-maps of the unit disc $\mathbb{D}$ with Denjoy-Wolff point $\tau\in \partial\mathbb{D}$. We study the rate of convergence of the trajectories of the semigroup to $\tau$, that is, given $z\in \overline{\mathbb{D}}$, we discuss the behavior of $|\varphi_{t}(z)-\tau|$ as $t$ goes to $+\infty$.
We also make a brief survey of the role of those semigroups in Loewner theory.