Cauchy-Fantappiè forms generalize the Cauchy-kernel [(1/2\(\pi\) i)(1/(\(\zeta\)-z))]d\(\zeta\) to n-variables. One possible application is, to derive criteria for holomorphic extendability of continuous functions, given on varieties. This development starts with \textit{E. L. Stout}, Duke Math. J. 42, 347-356 (1975; Zbl 0333.32003). In the last 15 years various generalizations are found: Let \(\Sigma\) be an oriented real hypersurface of class \({\mathcal C}^{\infty}\) on \({\mathbb{C}}^ n\). A function \(f\in {\mathcal L}^ 1_{loc}(\Sigma)\) is called a CR-function on \(\sum\), in case it satisfies the tangential Cauchy-Riemann equation in the weak form, that is \(\int_{\Sigma}f{\bar \partial}\alpha =0\) for every (n,n-2)-form \(\alpha\) of class \({\mathcal C}^ 1\) on an open neighbourhood of \(\Sigma\), such that \(\Sigma\) \(\cup \sup p(\alpha)\) is compact. Let \(D\subset {\mathbb{C}}^ n\), \(n\geq 2\), be a bounded domain with smooth boundary \(\partial D\) and \(E\subset \partial D\) be a closed subset of \(\partial D\). The author proves under appropriate conditions on D and E (see Theorem 3.1 for the details) a criterion for holomorphic extendability of smooth CR-functions f on \(\partial D\setminus E\) to a holomorphic function on D. The proof is based on the construction of appropriate Cauchy-Fantappiè forms.