The classical spaces $\ell^{p+}$, $1\leq p<\infty$, and $L^{p−}$, $1<p\leq\infty$, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ${\mathbb C}^{\mathbb N}$, $L^p_{loc}({\mathbb R}^+)$ for $1<p<\infty$ and $C({\mathbb R}^+)$, which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.