During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.