  Various properties of the (continuous) Ces\`aro operator $\mathsf{C}$, acting
on Banach and Fr\'echet spaces of continuous functions and $L^p$-spaces, are
investigated. For instance, the spectrum and point spectrum of $\mathsf{C}$ are
completely determined and a study of certain dynamics of $\mathsf{C}$ is
undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the
mean (and uniform mean) ergodic nature of $\mathsf{C}$ acting in the various
spaces is identified.
