  Let $\mu$ be a positive finite Borel measure on $[0,1).$ Ces\`aro-type
operators $C_{\mu}$ when acting on weighted spaces of holomorphic functions are
investigated. In the case of bounded holomorphic functions on the unit disc we
prove that $C_\mu$ is continuous if and only if it is compact. In the case of
weighted Banach spaces of holomorphic function defined by general weights, we
give sufficient and necessary conditions for the continuity and compactness.
For standard weights, we characterize the continuity and compactness on
classical growth Banach spaces of holomorphic functions. We also study the
point spectrum and the spectrum of $C_\mu$ on the space of holomorphic
functions on the disc, on the space of bounded holomorphic functions on the
disc, and on the classical growth Banach spaces of holomorphic functions. All
characterizations are given in terms of the sequence of moments
$(\mu_n)_{n\in\N_0}$. The continuity, compactness and spectrum of $C_\mu$
acting on Fr\'echet and (LB) Korenblum type spaces are also considered.
