  An investigation is made of the generalized Ces\`aro operators $C_t$, for
$t\in [0,1]$, when they act on the space $H(\mathbb{D})$ of holomorphic
functions on the open unit disc $\mathbb{D}$, on the Banach space $H^\infty$ of
bounded analytic functions and on the weighted Banach spaces $H_v^\infty$ and
$H_v^0$ with their sup-norms. Of particular interest are the continuity,
compactness, spectrum and point spectrum of $C_t$ as well as their linear
dynamics and mean ergodicity.
