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Let $\mu$ be a positive finite Borel measure on $[0,1).$ Ces\`aro-typeoperators $C_{\mu}$ when acting on weighted spaces of holomorphic functions areinvestigated. In the case of bounded holomorphic functions on the unit disc weprove that $C_\mu$ is continuous if and only if it is compact. In the case ofweighted Banach spaces of holomorphic function defined by general weights, wegive sufficient and necessary conditions for the continuity and compactness.For standard weights, we characterize the continuity and compactness onclassical growth Banach spaces of holomorphic functions. We also study thepoint spectrum and the spectrum of $C_\mu$ on the space of holomorphicfunctions on the disc, on the space of bounded holomorphic functions on thedisc, and on the classical growth Banach spaces of holomorphic functions. Allcharacterizations 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.

Ces\`aro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norm