<p>Using methods of the theory of the Laplace transform, we prove a theorem on the existence of a unique solution to an initial-value problem for a distributed-order differential equation in a Banach space, which involves a fractional Riemann–Liouville derivative and a bounded operator acting on the unknown function. We find this solution in the form of Dunford–Taylor-type integrals. The results obtained contribute to the theory of resolving operator families for equations in Banach spaces, including fractional-order differential equations and evolutionary integral equations; in particular, we generalize some results of the theory of semigroups of operators to the case of equations of distributed order. Abstract results for equations in Banach spaces are applied to a class of initial-boundary-value problems for distributed-order partial differential equations with polynomials in a self-adjoint elliptic differential operator with respect to the spatial variables.</p>