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We consider the generalized weighted Morrey spaces $\\mathcal{M}^{p(\\cdot),\\varphi}_{\\omega}(\\Omega)$ and the vanishing generalized weighted Morrey spaces $V\\mathcal{M}^{p(\\cdot),\\varphi}_{\\omega}(\\Omega)$ with variable exponent $p(x)$ and a general function $\\varphi(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\\mathcal{M}^{p(\\cdot),\\varphi}_{\\omega}(\\Omega)$ and $V\\mathcal{M}^{p(\\cdot),\\varphi}_{\\omega}(\\Omega)$. 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