Inspired by \textit{J. Peetre}'s abstract characterization of differential operators [Math. Scand. 8, 116-120 (1960; Zbl 0097.104)], the authors consider local operators \(T: {\mathcal F}_ 1(\Omega)\to {\mathcal F}_ 2(\Omega)\) where \(\Omega \subset {\mathbb{R}}^ N\) is open and \({\mathcal F}_ i(\Omega)\) are spaces of infinitely differentiable functions or distributions or ultradistributions. T is said to be local if supp\((T\phi)\subset \sup p\phi\) holds true for all \(\phi\in {\mathcal F}_ 1(\Omega)\). First the authors deduce continuity properties of local operators from results of their earlier article [Manuscripta Math. 32, 263-294 (1980; Zbl 0436.46005)], e.g., they show that T is always continuous if \({\mathcal F}_ 1(\Omega)\in \{{\mathcal D}^{(M_ p)}(\Omega),{\mathcal D}^{(M_ p)}(\Omega)\}\) and \({\mathcal F}_ 2(\Omega)\in \{{\mathcal L}^ 1_{loc}(\Omega){\mathcal D}^{(N_ p)}(\Omega),{\mathcal D}^{\quad (N_ p)}(\Omega)\}\), and that T acts continuously between \({\mathcal F}_ i(\Omega \setminus \Lambda)\), \(i=1,2\), for some countable subset \(\Lambda\) of \(\Omega\) in some other cases. They also give some counterexamples which show that the results are optimal. In the final section they characterize continuous local operators from \({\mathcal D}^{(M_ p)}(\Omega)\) or \({\mathcal D}^{(M_ p)}(\Omega)\) into certain spaces \({\mathcal F}_ 2(\Omega)\) consisting of ultradistributions as infinite-order differential operators \(\sum_{\beta \in N^ N_ 0}f_{\beta}\partial^{\beta}/\partial x\), where the coefficients \(f_{\beta}\in {\mathcal F}_ 2(\Omega)\) satisfy certain estimates ensuring the convergence of the series. Their proof is based on a representation of vector-valued ultradistributions by infinite series which generalizes the corresponding results derived by \textit{H. Komatsu} [J. Fac. Sci. Univ. Tokyo, Sect. IA 24, 607-628 (1977; Zbl 0385.46027)] for scalar values.