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We consider two $C_0$-semigroups $(e^{tA})_{t \ge 0}$ and $(e^{tB})_{t \ge 0}$ on function spaces (or, more generally, on Banach lattices) and analyse eventual domination between them in the sense that $|e^{tA}f| \le e^{tB}|f|$ for all sufficiently large times $t$. We characterise this behaviour and prove a number of theoretical results which complement earlier results given by Mugnolo and the second author in the special case where both semigroups are positive for large times. Moreover, we study the analogous question of whether the resolvent of $B$ eventually dominates the resolvent of $A$ close to the spectral bound of $B$. This is closely related to the so-called maximum and anti-maximum principles. In order to demonstrate how our results can be used, we include several applications to concrete differential operators. At the end of the paper, we demonstrate that eventual positivity of the resolvent of a semigroup generator is closely related to eventual positivity of the Cesàro means of the associated semigroup.