<p>In this paper, we study the questions of well-posedness of linear inverse problems for equations in Banach spaces with an integro-differential operator of the Riemann–Liouville type and a bounded operator at the unknown function. A criterion of well-posedness is found for a problem with a constant unknown parameter; in the case of a scalar convolution kernel in an integro-differential operator, this criterion is formulated as conditions for the characteristic function of the inverse problem not to vanish on the spectrum of a bounded operator. Sufficient well-posedness conditions are obtained for a linear inverse problem with a variable unknown parameter. Abstract results are used in studying a model inverse problem for a partial differential equation.</p>