Let \(f\) be a locally integrable function on \(\mathbb R^n\). Its Riesz potential is defined by \[ U_\alpha f(x)=\int_{\mathbb R^n}| x-y| ^{n-\alpha} f(y)\,dy,\quad x\in \mathbb R^n,\;0<\alpha<n. \] Consider also a positive continuous function \(p\) on \(\mathbb R^n\) and assume that \(f\) belongs to \(L^p\): \[ \int_{\mathbb R^n}| f(y)| ^{p(y)}\,dy<\infty. \] The authors study the problem of continuity of Riesz potentials of functions in Lebesgue spaces of variable exponent under the assumptions that \(p(y)\geq n/\alpha\) and that \(p\) satisfies a so called \(0\)-Hölder condition. They also study exponential integrability of Riesz potentials as extension of Trudinger’s exponential integrability.